This takes the long route to coming to the actual answer. Unfortunately, the ultimate answer is purely speculative. Whether any given entity would do something is all speculative compared to asking whether a given entity has done that thing. The best we can do is look at what has been done, what has been said, and use logic applied to the complexity and uncertainty of human behavior, and say, "based on the evidence this is the most likely answer." That's not an approach that will give us "the correct" answer, which is the typical StackExchange requirement, but I'll give it a go.
My approach will be to document what has been or is being done and draw the most likely conclusion based on that evidence.
Establishing an answer to whether the desire exists.
Evidence 1: Many Churches do pay for programs that help women in this situation. My own Church, with less than fifty adults old enough to work and earn money to give, partners with other area Churches to support a local organization called the Agape Center.
The Agape Center is founded, staffed, and financed by other like-minded Churches with no aid from anyone other than the members of these Churches. In essence, it is an extension of the Church for the Churches that have banded together to create this center.
The center offers information, counseling, support, and care for women with "crisis" pregnancies. We wish we could provide the type of financial aid you reference. We simply don't have the financial resources to do so, but we would if we could.
And we are not alone. Many Churches have similar outreaches. Some are better funded and are able to do more.
I honestly don't know of any Church or Church organization that wouldn't foot the bill to save lives, if it had the resources. I can't imagine a Church that would be so cold to those in need. It goes against basic, near-universal Christian principles of love and recognizing the value of life.
LDS, Catholic, Baptist, Lutheran, and many other Churches I've been involved with, attended, or otherwise dealt with have all had programs to help unfortunate women in this situation, and every single one would love to be able to do more.
Summing up the above, most Churches have a strong desire to do what they can to prevent and end abortions, and are willing to contribute financially.
The logical conclusion is, if they thought it were feasible I'm sure plenty of Churches would do it.
But that's pure speculation. So I'm marking this as a community wiki, expecting it to be deleted.