I prefer the Winchester Brass. I'm not sure if is real or perceived however. The Winchester alloy seems to hold up better, the vent holes seem to be more consistent and the annealing seems to be more consistent (less splitting). I do use both however because the Remington has been more readily available. I use Remington brass for my 220-grain bullets and Winchester for my 180-grain bullets.

I have been reloading Krag ammo since the mid 1980s and I use both also...  I will say that I think the Winchester brass cleans up much nicer than the Remington.  The Remington brass tends to be much darker and often displays more tarnish.  As for preference, it's a wash!

I was going to buy some brass and I wanted to know which is preferred, the Remington brass or the Winchester brass?

I prefer Winchester but RP is fine as well.  Whatever you buy, check the brass before loading it.  Many years (30+) ago, I got a box of new RP cases (components, not ammo)  where the flash holes were not finished properly; 50% had a thin layer of brass completely covering the flash hole & the rest nearly covering them.  All that was needed was to run a flash hole deburring tool through them.\ to clean them out


The barrel is pretty fouled with layers of carbon and my cleaning has uncovered some mild to moderate pitting about a foot below the muzzle.  Not sure if this should push me toward lead loads or stay with copper jacketing.  I do know you should strip all the copper out before going with cast. 

The fouling may not be copper.  Try soaking the bore for a few days with Ed's Red (available form Brownells or make your own - do a web search for the recipe)  Ed's Red is a superior solvent that dissolves and manner of crud.  I have a M1898 on order from the CMP with a barrel as you describe & will let it soak for a time before brushing it out.  I've used Ed's Red on a M1903 barrel that was clogged with debris from firing blank ammo.  It took several weeks as I changed the fluid 3x but it finally got it out.  At the end of the first week, a gray sludge poured out; the second week's drainage was thinner and the third week's fluid still was pink.

I've had horrible luck with the R-P.  Several neck and shoulder splits on the first firing of new brass, lot of 100.  It not a headspace or rifle issue as I've had on other problems.   I've got some old Rem-UMC and W-W Super that I've fired 10-15 times (light loads).