There is an Eastern Catholic on here who is also a Carmelite Brother, which seems to be working fine for him. Obviously, this also means that it can be done. I would imagine that the real question is whether it should be done, which probably depends on the individual.
The main thing would whether or not you are called to a particular spirituality which is found in a Western order. If so, then pursuit of a Lay/Third order might be something to consider. If not, then definitely not. Perhaps Brother (aka. ByzCath) will find this thread and share his thoughts.
Peace,