The ideal approach would be to simply allow your users to choose which region they sign into. If you don’t mind prompting your user for a region, I see no reason not to do this.
You could try against multiple regions as you suggested, but there isn’t an elegant way to implement it and there may be security implications (although nothing jumps out to me at this moment). I also can’t advise as to how it would comply with EU data laws or other data residency requirements, so that’s another issue you would need to consider.