If a 16x1.25 wouldn't thread in then a 3/8 BSPP wouldn't either, being .032" larger. Since the pitch difference of a 16x1.25 is less than .001" and the pitch dia. being only .004" larger I suspect you have a 5/8-18 thread because I would expect the metric bolt to go in more than just 2 turns. Anyway, try a 5/8-18 bolt if you have one easily at hand, maybe it will thread in easily. If it doesn't then the existing thread is very close to minimum dimensions, pitch dia. wise. And is probably 5/8-20 as you first stated. I wish I was close, I'm a retired self employed machinist and could easily determine exactly what you have.
Eric