I honestly don't think that concept is gonna work with 'normal' power levels. Might work fine in a custom high powered rifle, with a ridiculously low shot count, which means you have to shoot tethered indeed (we usually do it like that over here, on BR shoots).
No need for a carbon tank btw, steel scuba tanks are just fine, albeit heavier - anyway, that's clearly not the approach you want.
I tested the British Prometheus Piledriver pellets in .22 (well they were British, the rights and machinery were recently bought by H&N). They don't have the typical pointed bullet shape, as I guess that would be way too heavy, especially in the UK (12 foot pounds limit).
http://www.hn-sport.de/en/products/air-gun-pellets/hunting-and-field-target/piledriver.html Tested them in my old Evanix AR-6 - not the most accurate airgun mind you, yet it has more than enough power. Shot tethered to the tank, at some 190 bar. Wasn't impressed by their accuracy at all, there's better pellets in that weight category.
Just one test though, and there's obviously so many variables. When shooting bullets/slugs in stead of diabolo shaped pellets, high power levels are a requirement though. Not sure if a boat tail design can be beneficial, look at this guy shooting cans at a quarter mile using bullets with a flat tail, from his custom Talon/Condor hybrid:
Obviously, the rifling in your barrel is also a critical characteristic. Theoretically, bullets shot from barrels meant for pellets (which require lower twist rates) won't perform well at all.