LEE tumble lube bullets are not flat based, they are a bevel base and I've used gas checks on them. It takes an extra step to load as I put the gas check into the case first them placed the bullet on top of the gas check and seat it as normal. I've found spent bullets with the gas check still attached to the bottom of the slug.
This is just a bad idea. Let me explain. A gas check HAS to stay on the bullet. It's can't fall off without disturbing the flight of the bullet. Hence, the invention of the CRIMP-ON GAS CHECK. It makes certain the gas check stays on the bullet. Your idea, while sounding good, means the gas check may stay on the bullet, but most likely will come off.
Go back half a century, Lyman gas checks were simple cups that fit on the reduced diameter shanks of bullets designed to take a gas check. They simply slipped on, meaning they could and did slip off during the flight of the bullet. If it fell off straight back, without tipping as it came off, then no harm no foul. But tipping caused the bullet to get kicked sideways, causing it to wobble, sometimes to tumble,(if it was close to being under stabilized).
Now, a gas check is not a magic cure for leading. It allows a bit more velocity without causing leading that results from the flame cutting on the base of a bullet. It also acts like a scraper to the barrel, taking the big chunks of lead out with it. It prevents or helps in eliminating blow-by of powder gases that are another cause of leading.
Proper fit of the bullet to the bore is most important. To know what size the bullet should be, you have to slug the bore to know it's dimensions. Then size the bullets accordingly.
Bullet lube is the second most important item. A properly made bullet with the right lube can be pushed harder without leading.
