HACR breakers are made to handle those loads better than a "standard" breaker.
Lots of luck finding a Homeline "standard" breaker. The info page posted showed that breaker was HACR and SWD. That makes it a 'universal' replacement. Homeline likely doesn't have a broad line. ((That assumes your breakers are the same as the info page.)
Will be interesting if the contractor has an answer.
I don't see why the old breakers held and new ones don't.
If the transient time break is long enough that the compressor stops, starting a compressor against a high pressure differential takes a lot of current. If switching the A/C off and on you would wait a while for the pressure to equalize before turning it on. If I was desperate I might make something where a relay disconnects the A/C immediately then reconnects it after a time delay.
As I suggested the A/C may work on a 20A circuit. To try that, for many years 20A circuits have been in bathrooms, laundry and kitchen (including dining room).