The vintage English side by side guns ( mainly prior to WW1) with ‘damascus’ barrels had browned barrels, some American guns had a black or dark blue finish which can also show the pattern well but fades quicker. Actions, lockplates, butt plates, hammers and triggers were invariably case hardened, not primarily for colour or for show, but to harden the surface to protect the engraving and prevent wear. The trigger guard and top lever were usually the only blued parts although European guns often had that case hardened as well. An authentic restoration will duplicate the original finishes. Amateur or budget restorations frequently damage the engraving with over enthusiastic buffing and usually either leave the formerly case coloured parts in the white with either bright or satin finish or blue some or all of them. Case hardening is a specialized skill requiring specialized equipment and considerable knowledge and experience to avoid disaster, it’s a specialized sub trade in Britain.
I really hate to see formerly beautiful guns that have been tarted up to look pretty but stripped of the features that made them beautiful in the first place, they might be shiny but they’re ugly! In your case, trying to salvage enough of a formerly beautiful but now unusable and ugly wreck to become an inert wall decoration is a worthy goal in my opinion. You can never bring it back to it’s former glory and even if it were possible the expense would vastly outweigh the value of the finished product. By all means brown your barrels. It’s a slow, messy, repetitive job but the results, even by a dedicated amateur, can be very pleasing. And it can be stripped and redone until you are satisfied with the result. Getting brown is easy, getting it even over all is difficult, getting strong pattern is nearly an art. For your case hardened parts you will need to polish to remove pitting and scattered traces of case colours anyway, you can leave them white with whatever degree of finish you like or blue them, your choice. Be very careful to faithfully follow contours and keep flat surfaces true when polishing. You WILL damage engraving by polishing, try to minimize damage. An attractive combination is browned barrels, white action and lockplates with hammers, top lever, trigger guard, etc blued, but the choice is yours. Good luck and keep us updated as progress continues.