Is it that significant of an accuracy gain or what makes the .357 coated bullets that much better than .356? You know your stuff and shoot a lot, so I'm really curious about your interest.
James is the cast lead expert and can speak more on it than I can. But, basically, when it comes to lead, barrel fit super important. Most jacketed bullets are sized .355. Most lead and coated lead are .356. But you can shoot .356 jacketed bullets in 9mm Luger, and you can shoot .357 or even .358 lead in 9mm Luger. Many pistols benefit from using the larger sized bullet. They obturate better, and they're less likely to lead the barrel. With jacketed, both of my 9mm CZs are more accurate with .356, and in the very limited testing I've done with coated lead (with my Shadow Line only), .357 seems like it's trending the same way.
I haven't bought BBI in a few years, but they were the first coated bullet I used, and I liked the company a lot. If they're producing 9mm bullets at .357, that's definitely worth heading back their way for a trial. And by "trial" I mean I'll buy a case.

Note that this is their 125gr, though. When you go oversized, you will have to shorten OAL a few hundredths compared to standard size of the same profile, and in the case of 147gr bullets, that would likely have you seating past the mechanical limit where the interior of the case starts to narrow. If I were set on 147gr, I wouldn't buy more than 100 from BBI to test.
Generally speaking, ACME is my favorite coated lead company, though when it comes to 124/125gr, I have had great success with and love the Bayou 124gr TCG. And if YOU want 147gr, you might want to try out the ACME 145gr RN.