Having mech drivers also being commandos can work, I'm not sure many militaries would want to risk valuable mechwarrior doing jobs that a ground pounder could do though.
If one doesn't mind going that high/elite, there's the
DEST Commandos who are basically DCMS MechWarriors doing black ops for the Combine's intelligence agency. Given the many political schemes and enmities I could imagine some interesting missions both inside and outside the cockpit. I believe the Cappies have something similar in their Death Commandos?
The obvious downside being that it's a rather high point to start with (as Tai Dai Cultist said, humble origins tend to be more fun), and that the characters would have to be somewhat indoctrinated in order to having been picked for such a role. That doesn't mean they'd have to be automatons, but the players should be aware that it'd require extraordinary circumstances for them to do something like insubordination.
Ultimately, I think a ragtag mercenary company might be best-suited for your plans. Most of their missions would be 'Mech-based, but depending on how you design the campaign the characters could get entangled in all sorts of trouble even when "on foot", ranging from butting heads with locals to doubling as infiltrators in order to deal with circumstances where a BattleMech would be of little use. For example securing a mission objective inside a building. Proper merc units have their own infantry formations for such jobs, but when your team is too small to afford such backup or simply just started out and does not yet have recruited some ground hoggers, then they'd have to make do with themselves.
You could also set the campaign in the
Dark Age, which might help in that the DA sees a decline of technology and war materiel, meaning that (a) military units are smaller and (b) less well equipped, making it easier for a bunch of mercs to pull off some crazy stunts simply because they face less opposition/resistance, not to mention that you could tweak their 'Mechs efficiency (and thus the likelyhood of the players to use them) by keeping them low on ammunition and other critical resources, having them save up this stuff for when they absolutely need it.