I fish bead head nymph patterns all the time. A standard Clouser does not cast well, at least for me. Chris has a good suggestion in tying bead chain eye clousers. I also use quite a few flies that have a lead or brass wire wrap on the hook for just a little bit of weight without being as heavy as a bead, or lead head. I have cast small lead head jigs just to see what would happen, but didn't catch anything
Remember that many light-tackle rods are rated as to the size of lure they will effectively cast, and fly rods tend to follow a kind of size/weight progression. Tenkara rods are designed to cast fairly light flies, even if they have the backbone to fight a large-ish fish. The tip section in particular just doesn't have the spring-like force to really launch a heavy weighted fly.
If you're a purist then no, weighted flies are not ok. If you just want to catch fish, then anything the fish will bite and you can cast without injury is fine; but there is a limit to the weight from a casting standpoint
Just my 2cents worth
Bruce