The only way at this time to get a custom barrel is have another barrel rechambered and/or rebored, or have a stub done which is explained in the FAQs, this has been discussed a LOT in the past, try the search for more info, or look back a few pages, it was just discussed at length in the last month or so. GBO sponsor Wayne York of Oregunsmithing has done several rechambers and rebores for me, he also does barrel stubbing, plan on spending at least $300 for either plus the cost of the original barrel or donor stub and barrel.
http://home.earthlink.net/~oregunsmithing/The only advantage the 350 offers is it's a short action chambering which is a moot point in a Handi, the 35 Whelen will match or exceed the 350 so there's no point going there. I get 2900fps with the 225gr TSX in my 26" RMEF 35 Whelen, 2800fps is doable in the 22" barrel, same load with RL15. I suspect in the 1:18¾" twist 357 barrel it may do a little faster, but dunno about stability with the longer bullets, no one has tried one yet that I recollect. But if you're patient, the 35 Whelen may very well become an accessory barrel offering later this year, hopefully in a 26" barrel.
Tim
http://data.hodgdon.com/main_menu.asp