I’m contemplating a trip to Turkey in late May and early June with the possibilities being traveling from the Bursa area eastwards and north along the Black Sea coast as far as I can over about the 11 days (is Kars and Ararat reasonable?) or doing the south cutting down from Bursa to Fethiye and going to Antalya and ultimately to Antakya. Comparison on weather at that time of year? Accessibility due to the recent earthquake? Costs? Is the area close to Syria like Hatay reasonably ok or risky for a western tourist? I’d be driving back so really have only about 6 days out and 6 days to get back to Istanbul ultimately. Is the amazing museum pass deal still a thing? Any recommended accommodations in the major cities along either the northern or southern coasts?

I doubt I can be much help but, FWIW, my trip to Turkey was in May, and I thought it was a wonderful time to visit! Most of my month there was to the south and west -- mostly south of Ankara and from Cappadocia to the west. Lots of wildflowers, wonderful temperatures, etc. I doubt my old trip report would be of much use, but just in case:
Some Observations on a Fantastic Journey through Western Turkey

The best source I know for good climate data is https://weatherspark.com/

Sorry I can't provide better answers to your questions. I hope you have a great trip!

If you like rain, go to the Black Sea.
If you like sun and sea water temperatures around 22 degrees centigrade, go to the Mediterranean.

if your main purpose is to drive without visiting any of the stunning places you will pass through, a drive up to the Syriain or Armenian border and back is perfectly possible.
If not, forget it!
Bursa - Antalya with visits of Pergamon, Efes, Priene, Didim, Pamukkale, Kas, Demre, Olympos, Phaselis, Perge, Aspendos and Side would take at least a full week.

Thank you, this is very helpful especially the weather data. It seems northern Turkey is rather too cold in May, maybe better to visit in high summer.

Ive seen most of those things already last year but could easily take 3-4 days to cover some things I missed like Olympos, Phaselis and seeing things that were closed last time like the 7 sleepers of Ephesus and the archaeological museum in Fethiye. If I do do Turkey in May to June do you think lots of sites will be closed in June? I heard this is the time archaeology students get off at university and a lot of sites undergo excavation. It seems Ararat is not feasible but hopefully getting to Antakya and back is within reason.