I'm a big advocate of shooting off tripods, and having lugged more than my fair share of heavy lenses with heavy gimbal heads around as well I'm also a big fan of ball heads.

The real key here, in either case, is to have the rifle, particularly if it is heavy, mounted directly over its center of mass. This is why, on firearm chassis systems, you always see very long arca rails, so you can get that perfect position right over the ball head.
An Arca Swiss ball head is obviously the best, and you can get them used on ebay for just a couple hundred bucks and in a perfect world you'd put that on top of the highest weight rated Gitzo tripod, but the inexpensive ones you can buy on Amazon aren't terrible (and are what I use). An inexpensive tripod w/ ball head which is at least a serviceable way to get started can be had on Amazon for about 100$. The thing to keep in mind is that, what you're paying for in a tripod, aside from quality and brand, is light weight and increased rigidity. A big carbon Gitzo WILL be a more stable shooting platform than a 70$ Amazon Geekoto. So you can play the "buy once cry once" game, or you can try an entire rig for less than 100$ and see if that works for you before you go in with both feet. I will say that, if you're shooting from inside a vehicle, I wouldn't worry too super duper much about tripod rigidity, because cars are pretty squishy-wobbly things on their suspension, and one errant fart will make your crosshairs move a visible amount. Remember, in a car you're sitting atop four springs resting on balloons.
Regarding mounts, generally our airguns don't have picatinny rails, and when they do they're generally not over the center of mass. So you might want to go looking for a mount that'll give you a little more flexibility in that regard. Various flavors can be designed custom, bought on Amazon, or had for free. (I give a 3D printable design away free, just PM me if you want the file)
Speaking of cars, if you really want a car solution, another thing to consider might be a GROOFWIN, which is short for "ground-roof-window." It is an odd little device used by one of the photographers I've worked for, but if you want a variety of stable mounts on or around a vehicle, you might just check it out.
And, in case it wasn't obvious, I shoot off tripods all the time and absolutely love it. It is NOT as stable as shooting prone, bags front and back, but then again nothing else is. The tripod is a great way to get relatively stable shots from all sorts of odd situations and angles where you can't lay prone..... pretty much it does the exact same thing for rifles as it does photography as both require/desire the same thing.
I hope that helps.