I don't know of an engine that "just runs hot" enough to constantly trip the temp warning light. I also don't know what your dealer meant by telling you "they are not very common." The Ski-Doo 600 models in 2003 had a thermostat (part #420922517, $34.99), and they are readily available to order from Ski-Doo's parts website here http://parts.ski-doo.com/Index.aspx?s1=2gogqjfouqgkhaj6c2ceg1abe1&catalog_id=0&siteid=1
You'll also find drawings of the engine assembly on that website. A quick look at the cylinder, exhaust manifold, reed valve drawing will tell that you are correct in your assumed thermostat location under the head.
Are you sure everything is correct with how your cooling system is flowing? Often times an air bubble in the line can cause improper cooling and a temp warning light to trip in a sled that otherwise seems to run well. Also, I would consider draining the coolant and refilling with XPS Coolant (BRP part #219700362). You could also check your thermostat when you do this by removing it and placing it in hot water. If it opens up, then you don't have a problem with the thermostat. But I don't believe a thermostat issue should cause the temp warning light to go off.
Thank you for your answer.
If my heat exchangers are getting hot is the thermostat fine?
What is involved with pulling the head if I wanted to change stat?