Good ending cartridge, as well, as I move into my dotage. Have both a 38/55 & 38/40Rem barrel for my CPA. The 38/55 barrel is, as any good Texan should own, a nice lapped Shilen 12" twist barrel so no problem with 360gr+ bullets and 49gr of 2F Swiss. Another nice think about a 38/55 is it can be taken out to more powerful 38/50Rem at minimal cost if one decides they need additional power. Rick Moritz seems to shoot his quite well.
As others have stated, that crescent buttplate will be a problem. I started my BPCR career with one shooting a 200m all off-hand match. When I shot my first 'real' match it could not end soon enough with a very sore shoulder. Check on ebay for a proper Browning stock, as they will fit without much work. Another option for wood is CPA. They have a pattern machine for the Browning and can whittle you a shotgun buttplate stock with cheek-piece in short order.
Some longer-term problems with the Winroku............. twist is too slow, limiting bullet choice, and it is too light. Going to a 38/55 with the skinny barrel on the Winroku is sort of self-defeating. Lighter cartridge in a considerably lighter rifle results in a similar felt recoil to a heavier cartridge. And that trigger. From the factory the rifle will likely have a 5#+ trigger, or so, which can be frustrating, especially in off-hand. Lee Shaver used to sell sears that had been modified to get the pull down to about 3#, but they were not hardened and wore, so I always carried a spare. Easy enough to change out, but the Winroku action is difficult to work on and is nothing like a true High Wall, whether C. Sharps or an original. No parts are interchangeable between them if you need service. Even with Wayne's book they are still complex by comparison to John Browning's original and all the parts for them that are available (MVA, C. Sharps, DZ Arms, ASSRA forum, eBay, etc.). With the Winroku you are pretty much stuck with the factory.
The suggestion of a C. Sharps High Wall is a good one, as would be a CPA Stevens 44 1/2 with a 38/55 barrel. I up-graded to a C. Sharps about half-way thru my 2nd season of BPCR and shoot a CPA today. While many do well with the Browning models, I find the C.Sharps & CPA to be better suited rifles that do not require additional mods as do the Winroku or Browning. There are a number of good shooters that use the Winroku or Browning, but don't believe any of them shoot a stock rifle (trigger work or rebarreled or both).
About like another day I thought I had the match in the bag with 274 agg in position shooting and wound up in 4th place on x count. Four of us shot a 274 that day! I really got a kick out of those scores and me mentally counting my chickens before they hatched!