My answer below is geared more towards a general purpose vise - not specifically for firearms. A firearm's vise is very limited IMHO. You can get both of course, but for cleaning a rifle I use shooting rest bags that hold the rifle pretty firmly - good enough for cleaning and mounting scopes. I would bet you would use a general purpose vise way more for other purposes than just for firearms. I have also made up at least a half dozen types of "soft" and padded jaws for holding handguns without causing any damage.
A 4 inch Wilton, Irwin, Yost or even a Craftsman vise would fit the size bench you have well. Obviously not the strongest available but I do believe you are smart in considering the size and weight of your bench to pair up a vise with.
There are also lots of used vises available on ebay or at garage sales if you have the patience - lol. The vises I am referring above to are not expensive and would be adequate for smaller jobs. If and when you can get into a larger workbench, you can always upgrade your bench vise.
For a new high end American made vise these days (not many made here anymore), be prepared to take out a loan to buy as they are expensive!! That said, some of the higher quality Wilton vises made overseas are not terrible. Then again, for well under $75 bucks you can buy what I suggested above for your current bench.
I have a 5" American made Wilton Vise that I bought new the 1980's and it has served me very well for 45 years. It is a little small for my current 18 foot long & sturdy workbench, but these days I usually stick to smaller jobs so it still works well. I am always on the lookout for a used, heavy duty, larger, vintage, American made vise from yesteryear - and while I do not mind a light restoration project, I will not buy a "dog". So far - nothing has turned up locally and to pay hundreds to ship one is insane!
Pardon me if you have no other intentions for vise usage other than for rifle cleaning - then of course go with a specific vise system made for that purpose. I am sure there are a bunch of them on Amazon. Sorry for the long winded answer, but I spend many hours in my workshop and use my vises very few times for firearms so I wanted to pass that along.