The reason certain size breakers are in place is to make sure that too much electricity does not pass through a wire of a certain thickness. Passing too much electricity through can make the wire hot enough to start a fire. You can physically install a larger breaker, but electrical codes are there to protect you from fires, so just swapping in a larger breaker may void your insurance when an inspector sees that after a fire.
You may need to install a new, larger wire to the air conditioner; it is the safest course. (Or buy a slightly smaller, more efficient air conditioner that uses less electricity.) The 20-amp breaker in the other room may be too large to protect you, so that may need to be changed. Consult an electrician or two for the safest course of replacement.