As I understand it, a button ante is designed to speed things up by having only one person pay the ante for the whole table. I dislike that rule for a couple reasons:
1 - What if the SB busts out. Is the button forced to pay the ante twice?
2 - If the button is short stacked to not have enough to pay, the winner of the hand is cheated out of a full pot.
Ignoring those issues, the easiest way to think about the scenario presented is to think how it sould be handled in a standard ante game. In other words, a player with insufficient chips to cover the ante, is entitled to a portion of each player's ante. I.E. A main pot and side pot is created similarly to any scenario where a short stacked calls all-in for less.
So if the button has exactly the right amount for just the button ante, that's exactly how much he can win.
If he has less, then insufficient antes are collected, but that's still how much he can win.
If you're in middle position, and your stack is less than the amount of the button ante, that shouldn't exclude you from winning the ante. After all, the button ante is the pot before the cards are dealt. You can now call / bet for whatever you have and be treaded like any other short stack bet.
The notion that any player with chips can be eliminated by virtue of being a short stack, is wrong. I.E. If you have at least one chip, you get dealt in and are entitled to play for a small pot while the other players play on the side.