My main goal is to find a caliber that is relatively inexpensive to shoot and can effectively kill deer at close range.
Have you considered the .357 magnum??
Deer, 50yds and under? absolutely powerful enough. Ammo is cheaper than the .44 Mag (not sure how it compares to the .30 Russian short though, I don't have one, so I haven't paid attention to x39 ammo prices) Plus, you get to shoot .38 special for plinking. And, you're already versed in using a lever gun (.30-30).
Lever guns in .357 are short, light, and very handy. The Marlin takes an optic easily, though the stock isn't perfect for one it works well enough for me, anyway. A low power (3x is the high end for thick woods) scope or a dot sight, while untraditional looking on a lever gun, can be very effective.
Down side? The Marlin isn't common, or cheap. Prices I see on them are what I consider foolishly high. But, I'm still stuck in the past about the price of a lot of things. The last .357 Marlin I got was probably a decade ago a new looking pre-crossbolt safety gun that I gave $300 for. Don't think you'll find that price today though. sorry. Rossi makes a copy of the Winchester 92 that I've heard people speak highly of, but I've never had one, so I can't say from personal experience. The Winchester design is more difficult to scope, if that's important to you.
I was going to comment on the speed of a follow up shot, between lever and bolt action, something that might be important in a brush gun. For most people, the lever will be slightly faster. However, if you practice enough (and it doesn't need to all be live fire) you can get skilled enough with a good bolt gun to make the difference essentially insignificant.
My first deer rifle was a Remington 600 carbine in .308 Win, with half a box of ammo. Knowing I needed to shoot it to learn it, got me into handloading. I probably fired that rifle 500 times the summer before deer season, and practiced working the action, from the shoulder, at speed, a few thousand times, or so. That taught me a number of things. One of them being that a hot .308 in a 6lb rifle will beat the snot out of you, until you learn to handle it.
Another was that you could be pretty fast if you practiced enough.
We hunted the Adirondacks, mostly thick brush but 100yd shots could happen, and the one time I got one, my .308 carbine with a JC Higgins (store brand Weaver) 2.5x Post scope performed perfectly. If I had it to do over, I'd go with a lighter recoiling round, but I've grown old and wimpy since then.
Either of the choices you named would work well, which one are you more interested in trying out? Vs. cost.
A small reloading set up can still be done fairly cheaply, and can allow you to shoot twice as much or maybe even 3x as much (depending on caliber and load) compared to buying factory ammo. It opens up a whole new world.