The reason I suggested the 338-06 is because it's NOT a wildcat. But it's not a 375 either. I also vote on the 375/444, since JD says he will NOT rechamber a handi, doesn't like them I guess. He told me that in an email some time back.
I just emailed SSK 3 weeks ago and asked about having a Handi .357 barrel rechambered for the 358 JDJ. He said no problem, $100 for the rechamber. He did ask that I get from NEF an extractor for the .444 Marlin. He wasn't sure if he had any that would fit. For the time being I need to put that project on hold. But I suspect that by next summer I'll have one.