Hmm, you are right that Windows is required for updating the BIOS. If you do have means to try it, it would be great. There should be little risk about flashing the wrong BIOS - the flashing tool should prevent you from doing that. If you have doubts you could start at www.acer.com and click support, go through the whole process of selecting the model, just to make sure you end up at the right BIOS binary.
In response to your report we have tried two different Acer A315-21 with the latest Endless OS and they are working fine. However they have a CPU that differs from yours. We are attempting to get hold of a matching unit.