How do I refer to myself in a postdoc application research proposal?
You don't mention the formality of the application - for example is this an application to a specific PI or program, or an application for a fellowship to a nonprofit or government institution? 

I'd say the safest bet is to rely on extreme formality and refer to yourself as "the applicant." I see this all the time in proposals and use it myself when discussing prior work in grant applications. So I wouldn't characterize it as "forced," (inelegant might be a matter of opinion). 

However, if it's not very formal (i.e. submitting to an individual professor, or department/working group) then simply using first-person "my" or "I", etc., might be fine and suit your personal style better.