I would have 8 "conference champion" bids and 8 at-large bids. You have to seed the teams anyways to determine home field advantage so it's not that difficult to identify which conferences should get an auto-bid. Frankly, I'd grant the top 6 leagues auto-bids based on an algorithm from the strength of the conference over the prior 3 years and then grant the remaining 4 leagues access to the remaining 2 slots.

The last two "conference champion" slots could be allocated through one of two paths: (i) any two conferences can arrange a "play-in" game at season's end in lieu of a CC game, or (ii) in the absence of two play-in games among the four remaining conferences the auto-bids go to the two top ranked conference champions. You could give the conference champion qualifiers home-field in the first round and use seeding to determine the round 2 locations.

This is what I would do if I were the dictator of college football, but my post is probably not feasible because the P5 leagues will all want unconditional auto-bids and will balk at being so generous with auto-bids for the remaining leagues.