You can tumble lube traditional lube grooved bullets and successfully shoot them.
I cast and shoot 9mm/357/40 traditional lube bullets tumble lubed and a TL 45ACP bullet, they all work fine when tumble lubed.
I use Recluse's 45/45/10 formula of Alox/Johnson's Paste Wax/Oderless Mineral Spirits.
I put about 500 into a gallon Ziplock bag, pour the lube over the bullets and "massage" the entire bag. "Massaging" seems to get a better uniform coating and doesn't bang up the bullets nearly as much as tossing them against each other. In the summer months the lube is already very runny and easily spreads over the bullets. In the winter, I soak the lube container in a sink of hot water and the stuff transforms from a consistency of frosting to nonfat milk in a few minutes.
After the lube, I pour them onto a cutout cardboard box bottom lined with wax paper and after a half hour or so, roll them around. This helps spread the now tacky lube and prevents any "pooled" lube from building up on the side of the bullet. If I'm in a hurry, I'll have a fan blowing on the bullets and the stuff dries in about an hour. Otherwise overnight is plenty. If you use Oderless mineral spirits, you can leave the bullets indoors and the "stink" is minimal.
I've found that I only need to size my 40 cal bullets, I can load and shoot the other caliber bullets without sizing. If you need to size the bullets, you first lube, size, then lube again. Re-lubing can consist of putting the sized bullets into the Ziplock bag, squirting a teaspoon full of mineral spirits into the bag and massaging. There is already enough lube on them to recoat the sized surface. All you need is an light tan haze over the bullet, not a thick brown coat of "mule snot".
The 45/45/10 formula dries to a firm wax-like coating that does not attract dust and powder like straight Alox which never seems to truly dry. Even if you have to pull a bullet with a kinetic puller, you can simply brush off the powder from the bullet.
Here's the instructions for the 45/45/10. Good luck.