any UPS that can provide that kind of power would likely cost thousands of dollars.
Yep. You have to pay to play in this arena. 3600 watts for two or three minutes may not sound like much, but a battery-powered inverter drawing perhaps 5000 watts (to allow for conversion losses and other inefficiencies), powered from a bank of ten series-connected 12 V DC lead-acid storage batteries, is going to consume about 42 amperes for those two or three minutes. That's "only" about two ampere-hours of electricity (42 A x 3 min / 60 min/hour), so a bank of ten small car or motorcycle batteries should work fine. Don't forget to make allowances in your budget for a suitable charger.
But, see
@BobK's post #6 above. Maybe all you need is some upgraded convenience outlet wiring? Running a 10 AWG cable from a 20 A breaker to a new 20 A convenience outlet is probably going to be a lot less expensive solution than a battery-powered inverter.
The biggest problem I see is battery maintenance. Even with sealed lithium-ion batteries and charge-balancing chargers, this could turn into an expensive maintenance nightmare. But there are engineering companies that will design and build something for you, probably for a few thousand bux. If you want to DIY, plan on doing a lot of homework before purchasing anything. If it were my project, I would probably use sealed lead-acid (SLA) batteries, such as are commonly found in emergency lighting that comes on during a power failure. The installed life-time of an SLA battery is not great, especially if subjected to a deep discharge, but they are relatively inexpensive, and fairly simple to keep charged with a "float" charging system.
Buy, don't try to build, the inverter that changes 120 V DC into 120 V AC, 60 Hz, sinusoidal waveform with up to 30 A current capability. You may be able to find less expensive inverters from Asian sources,
such as are listed here, but
caveat emptor with that.