Short answer: no. The T3 action is neither strong enough nor long enough for the 338 LM. If you REALLY wanted to, you could do a 33 Nosler, but you'd be really limited in terms of bullet seating depth (read: the bullets would have to be seated such that they ate up a fair amount of case capacity).
Really, your best bet for a long range heavy hitter with the T3 action is the 300 PRC w/ Berger 230s or similar.