A transformer has several failure modes, all to do with the insulation.
1) It can punch through promptly, due to overvoltage. At these low operating voltages, this is only likely to happen if there's a very large mains spike, and there's not a lot you can do about it. Transformers designed for mains connection have to be designed to survive 1500 V spikes as a minimum.
2) It can degrade over time. Keeping the operating temperature down to something you can put your hand on will generally mean this process is slow enough not to be an issue in your lifetime, and is rarely an issue in low voltage transformers like this.
3) It can be damaged promptly with heat, aka smoking or bursting into flames. This is the usual way we damage transformers, when trying to push them beyond their design ratings.
It's not the current, or the current density that kills the transformer, but the temperature. As the temperature can take seconds or minutes to rise, this means you can safely draw excessive current from transformers for short periods of time, as long as the maximum insulation temperature is not exceeded. It also means that if you're going to do a soak test to estimate the maximum safe current, you need to run it for an hour or two, to let it reach its final temperature.
While it's fairly easy to measure temperature rise, it's not possible to know what grade of insulation is used in the transformer, premium transformers may use higher temperature capable insulation. Fortunately heat goes as current squared, so you don't lose too much possible power by making a guess. I'd be comfortable with a 50°C rise above a 25°C ambient, but if you took a chance and went for 75°C rise, you'd only be looking at sqrt(3/2) or roughly 20% more power.
Measure the resistance of a winding before you start. The tempco of copper is around 0.4% per C, or roughly 10% increase in resistance for 25°C rise in temperature. Run at power for a few minutes, disconnect and measure the resistance. Repeat the cycle until you decide the temperature has stopped changing, or you stop the test at that power because it's going to get too hot. This will allow you to determine the power rating of the transformer.
It has another rating, the regulation. On load, the output voltage of a transformer will sag by a few percent. You have to be happy that this sag is not excessive for your application, a typical figure is around 5% or so. This is rather quicker to measure. Even if you're happy with a large sag, you still have to operate the transformer within its power rating.
I've not discussed voltage overload. Don't. Unlike current overload, there is no margin for going a bit higher on the rated voltage. Don't.