The older 460 blows away my 1GB GeForce GTX 650 while the Radeon doesn't. which in my mind means a 460 is not minimum. Anyway, Deadlock is running fine so far; have to see larger battles.
I would love to pick up a Battletech game but that says, GTX 560 ti minimum. Graphs comparisions are not close...
On this machine I've got GTX 580. Runs all the games I have thrown at it including BattleTech, Doom (2016), and Project Cars 2 (2017). With that said, my preference is to play those games on my newest computer with Core i7-7700, GTX 1080 Ti, 16 GB of RAM, and SSD. That computer just happens to be in my brother's house where is a sizable room to move around with VR set. Reason for stated hardware requirements being what they are is because the games have been tested with that hardware and not the older ones.
In your case you could MAYBE get BattleTech to work with reduced resolution and visuals. You can look up used hardware, it cost less. If you still can't afford upgrade anytime soon, then it is better to save that money for the games that are more certain to work. Or save money for newer hardware and get games that cost nothing. To scratch BattleTech itch, I highly recommend MegaMek and MekHQ. You'll need at least Total Warfare rule book to play them though, and PDF version is available at least on DriveThruRPG for $14.99
You know what they say, don't you? About how us MechWarriors are the modern knights, how warfare has become civilized now that we have to abide by conventions and rules of war. Don't believe it. MekWars