Unlike the smaller calibers, .50 BMG plain ball is all mild steel core. If you want lead core you're getting into the more expensive rounds, hunting, match ,etc. So your criteria, for the .50 cal anyway, is a contradiction. You can have inexpensive plinking steel core or you can have more expensive lead core. There really isn't inexpensive lead core ammo for .50 like .223, .30 etc.
For example, surplus plain ball, steel core, is like 40 to 60 cents per .50 projectile, usually $2-$3 per round. If you want lead core, you're talking $2-$3 per projectile, $4-$6 per round, or more. Ads for .50 Ball ammo often won't say they're steel core because they're all steel core, it's kind of a given, like it being brass case.
Military surplus is the plinking ammo. Anything else is more $. Your examples of Barrett, American Eagle/Federal, PMC, etc are all mild steel core, military bullet. PMC does have a match load that is not steel core, but it is solid brass and about $70 for 10 rounds.
Not sure why your indoor range says you can't shoot lead, usually they say you HAVE to shoot lead, as opposed to steel core or steel jacket, but each range has its own rules. Most of the indoor ranges won't let you shoot .50 at all, regardless of bullet, unless they have a .50 rated bullet trap, which is considerably more expensive than standard installations. If they do have a .50 rated trap, it's usually a specific shooting lane and all .50 shooters have to use the one lane.
There's a very good reason why you don't find lead core bullets in .50 as commonly as you do the smaller calibers, why they chose steel instead of lead for that caliber, but I won't get into all that. It's a bit of a dissertation.