We have several teams in our league who have got themselves into serious cap issues (including one who has gone $104million over our $136million cap https://badcompany.myfootballnow.com/team/23).

This one gives me most concern though https://badcompany.myfootballnow.com/team/28

Honolulu Hurricane
Total Cap: $132,300,000
Dead Cap: $112,522,174
Room under cap: $-45,193,841
Need for Draft Picks: $14,447,042
Cap used next year: $50,481,366
Dead cap next year: $0
Players under contract: 14

I know that the AI will cut players to reduce the cap, but the total 'Save if released' amount on the 14 players is < $30million.

Will the AI be able to fix this?

If the AI cannot cut enough players to get under the salary cap, the team gets 25% cap space at the last minute. Just be aware that the first-round draft pick will instantly become the highest-paid player, and then be automatically cut.