Part of the problem is that we're using the word "saved"
We're burdening the idea of sanctification with the concepts of salvation and atonement. A creation of God is saved when that creation obeys the rules that apply to it (whether this is of necessity by the Atonement I leave to Jesus' good care, it's not relevant to me but for myself). We sometimes look at verses like Mosiah 27:30 (see also Luke 12:6) and assume that because our Redeemer cares about all of creation, that it means every molecule of that creation is "saved" and will be resurrected. But, what does this mean?
There is a remarkably complex scriputre found in D&C 93:30. It says,
All truth is independent in that sphere in which God has placed it, to act for itself, as all intelligence also; otherwise there is no existence.
If you read the verses surrounding this, you'll discover that the Lord is discussing the agency of humans, but that does not undermine His statement. To have existence, there must be intelligence. Intelligence allows the ability to act for one's self. Action requires a foundation of truth — a means of judgment — and that truth is independent in the sphere (context) God placed it.
Which is a wonderfully eloquent way of saying, the rules that apply to humans do not necessarily apply to animals, trees, bacteria, or any other aspect of mortal life.
We know the Earth will be sanctified
D&C 88:25-26 teaches us that the Earth (the globe) will be sanctified.
And again, verily I say unto you, the earth abideth the law of a celestial kingdom, for it filleth the measure of its creation, and transgresseth not the law — wherefore, it shall be sanctified; yea, notwithstanding it shall die, it shall be quickened again, and shall abide the power by which it is quickened, and the righteous shall inherit it.
Obviously, the planet doesn't have the capacity to make a choice. Therefore, describing it as "saved" makes no sense. Saved from what? What sin? What transgression? If anything, it's been a victim of Man's choices. Nevertheless, it must be subject to truth in its sphere (context) and it is successfully abiding by that truth. The Lord said that the Earth is fulfilling the measure of its creation and does not transgress the laws that apply to it, therefore it will be sanctified. How, then, does this apply to animals, especially given the example of Balaam's talking donkey?
We know something, but not everything, not by a long-shot. In fact, we basically know nothing at all.
Elder Gerald E. Jones in the August 1972 Ensign said, "The exact status of animals in the resurrected state is unknown except...."
That except is important both for understanding the condition of animals and for understanding our remarkable lack of understanding. We know that the Earth is doing its job and will be sanctified. We know that humans, through the atonement of Jesus, can be sanctified. It stands to reason that animals would enjoy something somewhere in between. But, that's a lot of mice....
We have several verses from D&C 77. This is lengthy, and I apologize. The emphasis is mine.
Q. What are we to understand by the four beasts, spoken of in the same verse?
A. They are figurative expressions, used by the Revelator, John, in describing heaven, the paradise of God, the happiness of man, and of beasts, and of creeping things, and of the fowls of the air; that which is spiritual being in the likeness of that which is temporal; and that which is temporal in the likeness of that which is spiritual; the spirit of man in the likeness of his person, as also the spirit of the beast, and every other creature which God has created.
Q. Are the four beasts limited to individual beasts, or do they represent classes or orders?
A. They are limited to four individual beasts, which were shown to John, to represent the glory of the classes of beings in their destined order or sphere of creation, in the enjoyment of their eternal felicity.
Q. What are we to understand by the eyes and wings, which the beasts had?
A. Their eyes are a representation of light and knowledge, that is, they are full of knowledge; and their wings are a representation of power, to move, to act, etc.
Frankly, that's it. That's all we've been given to understand. Does this prove animals will be resurrected? No, actually, it doesn't. It neither proves nor disproves the idea. But it does suggest that all of God's creations are important to Him. But these verses would be equally applicable if the schematics of creation were kept on a bookshelf in heaven (to use a mortal analogy) as to every animal being resurrected. The answer simply isn't clear.
And then comes Balaam's donkey. The donkey was given power to speak and she explained her story to Balaam and expressed her opinion. The real wonder of the story was that Balaam wasn't dumbstruck hearing a donkey speak — but we'll forgive the Biblical author as he was likely focusing on the moral of the story and not the specifics of history.
But if a donkey has the capacity to comprehend its surroundings, even if it doesn't have the particular ability to choose... Except we're talking about donkeys, which are notoriously stubborn, which implies the ability to choose. Remember, though, truth in the sphere of its creation. We don't know the rules governing donkeys, so for all I know, when a donkey is being stubborn it's fulfilling the measure of its creation (if only to be yet another plague for mortal man) and is wholly worthy of sanctification.
All this suggests — suggests — that animals (and possibly incects, the "creeping things") will be sanctified. But there are two wholly mortal perspectives that have not been addressed by the powers of heaven.
What would be the purpose of resurrecting all those animals, etc.? Humanity was given dominion over all things, and from the LDS perspective, we take the Lord's claim of inheriting all the Father has seriously. We believe that dominion will continue and extend to to the privilege of creation for those who enter the Celestial Kingdom.
And yet, why wouldn't we surround ourselves with the abundance of life? Of course, we don't really know anything about life after death, that's the reason for faith, but from a mortal perspective, the question is still interesting.
There is no official doctrine of the LDS Church that teaches animals will be resurrected or "saved." The idea of "saved" doesn't even make sense as it's a word that applies to mortal man and the rules mortal man must abide by. We have not been told what the rules are for other aspects of creation, only that other aspects of creation are abiding by them. We are therefore instructed that the Earth will be sanctified in its turn, which suggests animals can be sanctified, too, and it stands to reason that whatever God creates can, according to His will, abide in His presence. After all, He makes the rules.
But, resurrecting every ant would cover the earth miles deep in ants. The same would be true for mice. Despite Balaam's donkey, there are some impracticalities to even the idea of sanctification that underscore the fact that we are woefully lacking information.
However, if animals are resurrected and sanctified, LDS doctrine wholly supports that it would be through the power of Jesus Christ (whether it's a function of the Atonment or not is, at least to me, irrelevant. I'll let the Lord worry about how He does it).
Before I go, I'd like to point out (and my apologies to those people who are passionate about animals), why are we worrying about our favorite pet cat when we should be worrying about ourselves and our neighbors? The cat is fulfilling the measure of its creation and can take care of itself (and usually does).
Finally, with all due respect to Sandra Bradford Packer and the "Encyclopedia of Mormonism," the book "Teachings of the Prophet Joseph Smith" (abbreviated TPJS) is not scripture. There are examples in it of Joseph refining, modifying, and changing his early teachings as the Lord revealed more information later. Therefore, using it as a definitive source is inappropriate, and she should have known better.
To demonstrate this point, you quote from TPJS but you do not include the entire quote. Let me bring out the last paragraph (emphasis mine):
Again, there is no revelation to prove that things do not exist in heaven as I have set forth, nor yet to show that the beasts meant anything but beasts; and we never can comprehend the things of God and of heaven, but by revelation. We may spiritualize and express opinions to all eternity; but that is no authority.
TPJS does not tell us when Joseph made these remarks, but it was likely before the revelation of D&C 77. Why? Because in that revelation we learn that the beasts of Revelation (which Joseph is talking about) "are figurative expressions."