Yes, they are susceptible to temperature. Seasonal temperature changes are mostly what causes them in the first place. Additionally, researchers have used water temperature to create otolith barcodes in captive fish before release. Otolith bands are the result of seasonal changes in growth rate, which is highly trusted to temperature and is why it's a less reliable technique in tropical or deep water species.

That getting said, climate change and other things are unlikely to overwhelm the seasonal cycles. Yes, it's gotten warmer, but it's still colder in winter.

It's also preferable to have validation for a given taxa, but i wouldn't expect that particular worry to be the failure more.