WHERE VACATIONS START!
A requirement of residency in Panama is obtaining a Panamanian DL whether it be temporary or permanent. This was explained to us by our attorney, Sertracen and by reading the law from Migracion. Sertracen is the agency responsible for issuing DL's. Maybe you should stop by there and ask if this has changed recently. We had to renew ours 6 times before we got the four year license.
Setracen and Migracion are next door to each other in the shopping plaza next to Pricemart. There are English speakers at both and would certainly be worth an hour to check as this could cause problems with your residency and could possibly get your car confiscated.
90 days is correct. It seems like it would be easier to go to the embassy and get your license apostle and then go to Sertracen and get your Panamanian license. You have to do it anyway within 90 days if you hold a temporary visa. A recently stamped passport will not work if you have a temporary residency. You have 90 days from the time your temporary carnet was issued not the last time you entered the country.