I think I’ve heard of the international driving permit being stamped for a year. Maybe different DMVs have different policies. I really don’t know whether you could renew it after 3 months, but I’m pretty sure that you couldn’t extend the validity past the expiry of the IDP itself.
[Edit: for up-to-date info on IDPs, see this thread:
[Foreign-issued International Driving Licenses/Permits ]
I have to renew my Taiwanese motorcycle license when I renew my ARC. I would say it’s worth doing the test and getting getting a Taiwanese licence if you’re going to be here a while, especially if they’re only going to stamp your IDP for 3 months.