The rings/bores may well have been the problem but if you're running in on fully synthetic the new rings won't bed in so you're back where you were. As Mark said, you need a running in oil or a semi/mineral oil.
If there is no external leak then you have to be burning it (or losing it into the cooling system via cracked block, bad gasket etc but you'd see that in the coolant), if it's not coming past a valve guide it has to be passing by the rings (assuming you don't have a cracked/holed piston). Maybe suggest to them that you take it to a Z specialist and they foot the bill?
They don't have to smoke to burn oil. Mine burns loads but it's only visible under hard use where as when I just had a race cat it left huge clouds everytime I got on the throttle, especially after any engine braking. The stock cats burn virtually all evidence of it.