Could an uninterruptible power supply for energy intensive appliances balance the power draw and help my breaker stop tripping?
Home UPS aren't designed for peak-shaving.  Any system that could do that would have to know what the load is on the entire circuit (i.e. what the rest of the loads are drawing) and then try to supply additional power onto an active bus...you see where this is going.  Much easier and very likely cheaper to put in an additional circuit.  Do not attempt to replace the breaker with a larger one as this increases the chance your wiring will catch fire.