I'm in with Harold on this one.
Your 15A breaker most likely feeds 14 gauge wire. For a 20A breaker, you need 12 gauge wire.
If you change the breaker without changing the wire, you could end up with a housefire. as a matter of fact, you'll likely end up with a house fire.
Your best bet would be to have someone add an additional 20A circuit dedicated to the AC unit. Use the properly sized wire, AND use a 20A, not a 15A outlet, at the end of the run.
If you have basement access it's not terribly difficult for an electrician to add a circuit like this to a first floor room.
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