We can sense the primary current of this transformer, with a shunt and an isolated delta-sigma ADC. The FPGA squares the samples and filters, so we can pick that up and square root to get RMS current. The tranny is rated for 240 VA, which would be 9.6 amps RMS in the primary.
So I ran it for a few hours with 10 amps DC in the primary. Temp rise was about 26C in free air. I think people design transformers for equal copper loss in the primary and secondary, so temp rise would double when loaded in the system. But wafting a little air over it cuts the rise in half or so, so I'm back to something like 25 c rise in real life, where there will be lots of air. So we'll set the software shutdown at 12 amps maybe.