You are 4cm longer inseam than me for same reach. Personally, if you really want a fixed wheel, I’d get a 56 cm PW. The saddle to bar drop need not be so extreme (mine is typically 6-8cm and you would be 10-12. THEN I’d flip the stem. I know it’s not the best looking solution compared with my Thomson -17, but needs must. And you should have a rear brake. Speed modulation (as opposed to braking is not just a front wheel activity).
It’s such a great bike, that youc an cut your teeth riding fixed, then once you’ve really caught the bug, go custom. If you don’t like it, nothing lost. I ride my customised PW about 100 miles a week.