seeking help identifying small buzzing transformer from my UPS, or advice on how to silence it
You have given no indication that the transformer is broken, and that replacing it will therefore do anything useful.

Options are:

Live with it.

Get a different UPS.

Muffle the whole UPS somehow, keeping in mind its need for ventilation.



Adding foam to the board, or doing anything else inside the unit is a bad idea.  You don't know what each part does, and what thermal considerations it has.  You shouldn't be inside something with dangerous voltages without really understanding it anyway.

Perhaps you can put the whole UPS inside a cardboard box with sufficient vent holes to allow air flow.  The holes can be arranged to muffle sound in the direction you care about.  If you do this, at least at first, put a thermometer in there and monitor the ambient temperature the UPS will now see.  Check the UPS datasheet and make sure you're not exceeding its requirements.  You need to check this under sustained maximum load, both while operating from the line and from the battery.

I'd probably just get a different UPS and use the first one in a different place where noise is less of a issue, or sell it, or something.