The law for military service exempts you for only 1 year upon getting HHR (as a naturalised citizen)
Could you therefore, get Taiwan HHR and then leave Taiwan in the first year and return after service age?
Yes. Many have done so. I used to send my immigrant clients sons overseas first on student visa’s so once they got PR they did not return until they were exempt. They could return for visiting and not be drafted.
Do not not get stopped from leaving the day you get HHR and ID card.
See if “overseas compatriot endorsement” applies to you, I guess