I would be very careful about increasing the amps on any circuit breaker.
That is not to say doing that is always a bad thing, especially if the one installed was under rated for the job. But if a 350 amp circuit breaker keeps tripping, that is a lot of current draw and I would want to be certain there is not something wrong that could cause a fire before replacing it.
A typical air conditioner does pull a lot of amps on startup, but these are momentary loads, and they quickly settle down to around 13 amps at 120 Volts, or about 130 amps at 12 volts. A circuit breaker should handle the high starting loads without tripping.