The Lyman is the more "historic" of the lot, though not a truly historic gun. The Thompson Center Hawken is "Hawken-ish", while the Renegade is even less so with the shotgun style butt plate. Both are good guns. The Renegade is a great huntng gun, shorter than the others, and feels more like a modern hunting gun/shotgun.
Another great option, and they can be found is the Thompson Center New Englander. I have one with 3 barrels (.50, .54 and 12 gauge), though it's not for sale. I don't know that there is a better sidelock avaialbe for a hunter. They are getting more difficult to find, but pop up on here from time to time. While this also is not a historic gun, it's a great starter gun with all of the "feel" of a traditional sidelock.
If you jump into this sport, take it from me, you'll eventually move past the production guns (T/C, Lyman, etc) and will end up falling down the custom/antique gun rabbit hole. I own 2 T/C guns, and outside of buying more to flip them, I doubt I'll buy another to keep them. Once you see the beauty and art in the custom and antique guns, the production guns lose a lot of their luster
However, and this is coming from someone who still considers himself a "beginner", the best way to develop a passion for traditional muzzleloaders is to get one and shoot it, regardless of whether it's a beat up old production gun or a handmade custom.