You could consider the DANCO Model 10672X Hydroseat. It may fit your needs. Read up on it. You put a normal wax ring under it. I used to think you put another ring on top, but I see they are planning that you just put the toilet on top with no top wax. This would fit cases where the feet are going to be on top of the finished floor, and you can drill holes for screwing down the feet. You can alternatively use the holes not on the feet to hold it down. That way you don't have to drill your hard flooring. You don't have to be as careful with the shims, because adding shims does not stretch the wax.
A closet flange holds down the toilet. I am not sure that your flange can reliably do that. You can put a repair ring on the floor above. Pack in enough wax, with the top ring sticking up at least 3/4 inch.
PASCO 21013, Jones Stephens C85000, Oatey 42775, and Superior 21015 are repair rings with mounting tabs outside. These sit on the finished floor, and screws hold the repair ring down. The screws can go through the repair ring legs (requires drilling the tile) or can go through the holes that are more inside.