Both the 338 Federal and the 358 Win are great short action cartridges that would be good options to pair with your others on your Bighorn rifle.
Bullet weights from 160-225 gr between the two with enough punch on any game in north america at short to medium ranges (dependent upon target animal).
Have used both of these on deer, caribou, moose and elk to date, and seen the 358 used on bison, and read many good stories on their use on grizzly and brown bears over the years.
Out to 250 yards, they carry good energy and require a dead on hold when zeroed at 200 yards and still stay inside a 4" circle, even from a 20" barrels.
I have primarily used the 200 and 210gr factory ammunition in both, and handloads in the 358 to date with 220 and 225 gr bullets.
I haven't seen the bighorn actions either so cannot say anything about them. Glad to hear reports here that they are very smooth!
The Defiant actions seem smooth, but I was very impressed with how smooth the Borden actions were, when RMR was building customs with them.
My one build with a Benchmark barrel (6.5 CM with the 5R rifling in a 1:8 twist) shoots very well (1/2 MOA with factory ammunition).
My current build is on a LH Sako 85 Finnlight II to 338 Federal with a 22" carbon fibre barrel 1:10 twist in a light contour. (love how smooth these factory actions run) Planning on using the 200gr AccuBond in this rifle.
Will see how the factory stock works for now. Not sure as to what it will end up in, but probably also have a nice piece of wood (French or New Zealand) for nice weather hunts.