The question "how can God be allowed to deliberately take a good through an evil?" appears to muddy the waters. It is one thing to bring a good out of another's evil act, and another to commit an evil act yourself for the sake of a good end.
For instance, imagine you are an undercover officer, and you must stand by while a member of the criminal organization you have infiltrated murders an innocent person. You stand by so that later you might have the evidence required to bring the whole organization down and stop all future crime. The individual murder is a bad event, and you do have the power to stop it. But, you can bring about a greater good by permitting the evil to happen. It is not immoral to do so.
Contrast this with an officer who goes under cover, and, in the process of his undercover work must murder an innocent person. He is not permitted under double effect to commit a murder so that he may bring the eventual good of the toppling of the organization.
In the first example, the actions of the officer are morally neutral. He does not get involved in the murder. He is a bystander. In the second example, his actions are morally evil. he is murdering someone. This is the distinction we generally try to make with double effect. You can know an evil outcome, a murder, will result from your inaction. Or, in other cases, from your action, say, an innocent bystander might be caught up in cross fire when you go to war with a tyrant like Hitler. You cannot do an evil thing because you want to obtain a good end. EG, you cannot steal from honest, hardworking people who have a bit of excess to their wealth in order to give that wealth to people who need it (this doesn't mean social welfare is wrong; governments actually have a right and indeed a duty to levy reasonable taxes and to allocate funds to the common good, what I'm talking about here is Robinhood style vigilantism).
Once we have understood what double effect really says here, we can see that God's permitting evil is more like the first undercover officer. He does not commit evil acts Himself, but He will permit free willed humans to make evil decisions and to commit evil acts.
The Doctrine of Double Effect: SEP