In the text Mr. Bennet says of Mr. Collins that "he never saw in the whole course of my life". Further indicating they were distant cousins..for fun how about this?

The property originally belonged to Mr. Bennet's Great Grandfather and that Mr. Bennet is the last direct male in that line. Maybe his father and grandfather also had more daughters than sons. So then it goes back to the next male sibling to the Grandfather. And saying that Mr. Collins is the Great Great Grandson of the originator and has outlived his father and grandfathers.

Looking at my cousin chart--Mr Collins would then be Mr. Bennets 2nd cousin 1 removed.

I agree with you on the name difference and possible change. There is another way that it could have been done but based on Jane's family situation I too think it was more likely meant to be adoption.