Discussions cannot be had without definitions
The term "cowboy" is more ambiguous than "assault weapon."
If you want to be a cowboy, go be one, whatever you define it as, go do it. Be what you want to be (this world would be better off with a few more cowboys, personally).
If you want to LOOK like a cowboy, go hook up with some CAS guys and gals. They have the look and feel of yesteryear down to a science.
But it seems your original post was more along the lines of: "What gun would you suggest that will make me feel like a cowboy, be fun to shoot, and be worth the money." I could be wrong, but that's the vibe I get from your OP.
Since you said you're not into CAS, you probably don't have any need to invest in the more expensive, competitive, high-performance CAS approved brand names like Winchester, Henry, or Cimarron. While these firearms are great and attempt to celebrate heritage only after being performance ready, they are probably an unnecessarily high price for the non-competitor.
If you don't mind a smaller caliber, go invest in a Marlin or Rossi .22 for your plinking pleasure. Plus, they're excellent get-rid-of-the-garden-pests rifles.
If you want a big bore, but to not break the bank, Rossi and Marlin also make affordable lever guns in the bigger calibers (I.e. 45LC and 357 mag).
My overall suggestion, go with Marlin. They are a standard favorite among CAS competition shooters and recreational trigger pullers. They are reliable, versatile, affordable, diverse (most major calibers), and popular (the advantage there is that you'll never have a shortage of advice, replacement parts, or experienced shooters).
The first question I ask anyone when they ask for weapon purchasing advice is this: why do you think you want this or that kind of weapon?
I'm not trying to invalidate your question at all, but weapons are a financial investment at almost any level, and aside from being dangerous, firearms should not be purchased lightly.
If you know someone who has a lever gun, go offer to buy them a beer if they'll let you shoot theirs.
Make sure the beer comes AFTER the shooting (be safe!).
Best of luck.