How should I refer to PIs in a statement of purpose (US)?
tl;dr. In a formal situation, be formal, otherwise...

The US is pretty informal about titles (unless the professor is from a German university). So either should do in general (see below). Even an assistant professor can be referred to as "professor". It is even possible, if rare, that an Associate Professor won't actually hold a doctorate. But no one is likely to be offended. 

You might, however, after a first usage, be told that some other thing is more appropriate. Then you should just do that. 

My own doctoral advisor did have a Germainc type doctorate and additional beyond doctorate degrees. But he was happy enough in the US to be referred to with any reasonable designation. I also had a colleague who was at least an Associate Professor but held no doctorate. He did, however, participate in very important theoretical research - with Edsger Dijkstra, actually. 

But this is how you refer to a person in person or in email and such. For a more formal designation, such as in a paper, you may need to refer to an actual title. "Associate Professor Smythe" or such. The context you give, suggests a formal title is called for - an application.