How bad is it to momentarily overload a transformer?
A transformer should not be overloaded for voltage, at all, for any time at all, at all.
A transformer can be overloaded for current, as long as the temperature of the windings stays within the limits of their insulation.
How long you can overload it depends on the thermal time constant. A 1 kW transformer is a pretty large beast, and I would guess that the relatively small overload you're subjecting it to could be tolerated for many 10s of minutes.
If you want to see what temperature the windings are getting to, and have a DMM, then you can make use of copper's (well, pretty much all elemental metals) temperature coefficient of resistance of 0.4% per °C. Unfortunately, you don't know what the insulation rating of the transformer is, so this exercise could well be moot.
Measure the resistance of the mains winding at ambient. Run it for an hour or three at 1 kW to reach thermal equilibrium, and remeasure the mains winding resistance. Estimate the temperature rise using the 0.4% temperature coefficient. This rise is what the transformer has been rated for.
Now run the transformer at 1250 W, measuring the resistance/temperature every 5 or 10 minutes. Stop the experiment when the temperature rise threatens to exceed the rated rise. Now you know how long you can over-run it without exceeding its rated temperature rise.