How to determine the max current my USB Charger can safely supply
Its hard to say what current a cheap power supply from aliexpress can supply safely. But if the rating says 2A and you only need 1A it should be fine. From your description i assume that the device is not getting too hot to touch for a longer time, which adds to the assumption that 1A load current should be fine. If you want to try a somewhat experiemental approach you can stress the device with say 1.8A (Keep an eye on the output voltage aswell) and monitor its case temperature over a longer duration (until thermal equilibrium is reached). If you are still below or equal 70°C (@20°C ambient temperature) you are totally fine (only valid if the case is non metallic). Have a look at IEC 60950 if you want more detailed Information about how hot the surface of electronic equipment housing is allowed to get.