Since you are shooting a single shot you can use bullets that are longer, heavier and more pointy than permitted in a lever action.You will want a velocity of about 1100 FPS to about 1400FPS.
I would try 5744, SR4759 and IMR 4227 if you have any of these. SR4759 is probably the best but it was discontinued about 3 years ago. If you are going to shoot serious groups I would get a Postel or Creedmore profile bullet of about 300 to 320 grains. Since you can afford the nice Browning/Winchester rifle buy an equivalent quality mold from Steve Brooks or Buffalo arms for it. Buffalo Arms also sells some ready made bullets for you to play with before you have to get serious about a mold.
https://www.buffaloarms.com/reloadi...t-bullet-20-1-alloy-spg-lube-box-of-50-379320
If your rifle has a chamber mouth of at least .398 or so you can use WW brass. If it is any smaller you may have to use the thin Starline cases. The WW cases are about .009 thick and the Starline are .0075 or so. With a groove dia plus .002 bullet you may not be able to chamber a WW case.
The 38-55 has about the worst variation in groove diameters across all the different rifles ever manufactured.
It was originally supposed to be a real .38 cal with a .379 to .380 groove diameter. But they have been manufactured at .375 up to about .383 groove diameters. The sorry SAAMI drawing permits the chamber to be about .394 maximum which means that some companies have manufactured rifles with small chambers and large bores that are not really compatible with shooting groove diameter or larger grease groove bullets.
You can check the bullet length vs barrel twist here and at several other sites.
http://www.geoffrey-kolbe.com/barrel_twist.htm
There was a Rifle or Handloader Magazine published about 8 to 10 years ago with a test of the Browning/Winchester .38-55 by Mike Venturino. I think the groove diam was .379 and he got really excellent accuracy from the test rifle with case bullets.
Finally go over and join the ASSRA (American Single Shot Rifle Association) Forums so you can ask questions there.
They have a lot of very experienced single shot rifle shooters there and the 38-55 is one of the top 4 or 5 calibers they shoot.