In my experience, the lack of a PhD won't be an issue; the lack of a lab science PI may be. If you want to do clinical trials, that generally isn't the type of person programs are looking for to fill the PSTP spot. If you want to do lab science, you'll have to convince them of your desire/aptitude and will probably get quizzed heavily on your second author pub, unpublished/ongoing research, and what your role was on those other basic science pubs.
There are also programs where you can get a PhD during fellowship or grants specifically supporting non-PhDs doing lab work during fellowship; if available at the places you're looking at, you may want to lean into those to get the training you don't have.