Yes, sea level should be conditional on observed sea level every year, but as it stands, it only takes a few metrics, such as over SLR during the 20th century, then present-day rate, and indeed some projections to 2100. The reason is to avoid having to deal with complex auto-correlation for now. I was quite happy with the model until recently but I am currently not because of the divergences and shows some variation from one chain to another in the tails (this can be reduced with target-accept, but still). I’m happy to find a formulation without Potential, and am trying to follow your logic, but I’m also struggling to come the other way. Here what you are saying is that I should use a different model 
?! What if my quadratic formulation had some physical basic, and I simply wanted to have my parameters distributed in such a way that my outcome is log-normal ? I mean, it’s not like the quadratic relationship between SLR and temperature is established (it’s not and indeed can be changed), but I’d think an exponential of a quadratic would be further remote from reality and it would be hard to find a physical justification for such a form.