Who would need armor piercing rounds anyway other then the military or law enforcment or anyone intending to pierce armour. Here in AZ a local gun store has tons of old surplus ammo mostly steel core i guess is classified as AP ammo but its sold dirt cheap since most of it is corrosive or mildly corrosive I don't mind using mildly corrosive ammo when I'm done shooting I just swab the barrel with some hoppes number 9 and spray the bolt with rem oil. They sell some czech 7.62x54 Rimmed ( Not russian) from the 60's which is mild steel core copper jacket which when originally manufactured was never intended to be AP but the "EXPERTS" with the BATF classified it as AP.
But anyway the stuff shoots great out of my mosin nagant prints 2 inch groups at 100 yards with open sights the best part is its like 2 dollars a box of 20

Yeah its legal to own, sell, and shoot. but by law the retailer has to keep record of the sales on it. But aslong as you don't intend to do anything illegal with it you have no worries. untill the BATF decides to ban it altogether. But It is illegal to manufacture, import or export.

With regards to Federal Law, AP rounds, as defined to be regulated, are specified as Hand Gun bullets. Rifle rounds are exempt, with two exceptions, those being .308 and 7.62x39mm. Someone made a hand gun to shoot them. State laws may ban AP rifle rounds, Conneticut banned them not ong ago.

Many .50 BMG shooters shoot AP because it is cheap (relatively) and it is among the most accurate milsurp projectiles available. The hardened steel cores are lathe turned. Most of it is old, before Tungsten was used.

A solid steel projectile is possible, but highly impractical. In the early days of .50 BMG competition, many competitors shot monolithic solids, CNC turned from a steel/lead alloy (97.5%/2.5% if I recall correctly),called leadloy steel. The small lead content is needed to prevent galling and the alloy machines real nice, too. Most monolithic solids in .50 BMG are now made from bronze, or similar.

Not only that, the bullets are slightly sub-caliber (.502"), except for a driving band at the rear(.511"), a cannon projectile type design. This type of projectile is called a borerider. A borerider chamber has a 2nd throat. The projectile slip fits into this throat when loaded into the rifle. The actual bearing surface is the small driving band at the rear. The forward part of the bullet shank behind the ogive "rides" the bore, with only .001" engraving, compared to the usual .005"(.50 BMG)

Any projectile that you get turned in any volume will need to be done on a CNC machine. To make it worth it, we are talking a big volume. You better have a lot of barrels. You will need them. Unless your barrels are reamed with a 2nd borerider throat, your projectiles will need another driving band up front called a bourrlet. How wide the driving bands? How to load for such a round so it will not blow up? I have not got the slightest idea.