A UPS with a battery?
There is NO need for isolation. The computer runs right off the line and the battery is in a sealed box, can float at line potential. (Things are different in very-large UPS racks where batteries can be changed "hot".)
The transformer does two things:
Steps 120V down to about 12V at low current to constantly charge the battery.
Steps ~~~12V AC (from battery and inverter) up to ~~~120V at high current to run the computer for a short time in a blackout.
The ratios must change from standby to Run mode. We need 120V down to more than 12V to charge. In Run, the battery sags below 10V but the output must stay near 120V.
I have tried to re-purpose these things and decided the designers were much too clever for me. Between Best and APC, brutal competition drove the price of UPSes _VERY_ low, which means they are full of penny-shaving tricks. Add to that the many different ways inverters can drive transformers, and the infinite variety of Run waveshapes possible.
For more fun: small UPSes do not deliver sine-waves in Run. One I had made pure square waves at 150V peaks. This runs electronics just fine (similar to the 168V peak of a 120V sine) but makes a lamp Really Bright because it feels 150V. Most now use a 3-step wave which has peak/RMS ratio more similar to a sine, but is far from a sine. There are also 5-step waves, and whatever waves heavily filtered to near-sine, but you won't see these.
> you have reached ~1.5 times the voltage to which the transformer is rated
UPSes have a special case. Charge can last forever, but the Run time is known exactly and is very short. Office-size UPSes give you much less than 20 minutes of run time, often less than 10 minutes. It can NOT run longer because you only paid for so-much battery! Also most UPSes will only run their full 7 minutes once or twice in the 5-year life of the battery, and most USPes are discarded when the battery gets old (most offices don't have a screwdriver; a whole new UPS costs very little more than the battery).
If the original wall-wiring were known.... but I have tried to trace this through the PCB through filters and relays and it is a real puzzle. We know that two leads go to the 120V/240V wall, for Charge mode, and this is safe "forever". However there may be step-up leads at higher voltage for Run mode, not good for more than a few minutes at the higher voltage.
And it is very likely it has NO isolation. The only use I can think of is charging a battery in a sealed box or running 12V lamps from wall-power.
> It's PCB had a burnt out trace.
That can be patched. However that is not the problem, and if you do a large-wire patch you will probably have a bigger problem. Most likely a diode or an inverter transistor failed short. The entire battery capacity flowed into the short in an instant. While thee may be fuses, an inverter has too many ways to go wrong to cover every possibility The PCB trace widths are designed so they will blow fairly quickly when stuff goes wrong. Throw down a patch of house-wire, it will hang-on until something else goes boom.