I have done a lot of those big beautiful 'Bugs'. They are almost impossible to 'smoke' in the normal sense of the word. When we say smoke, we usually think of smoke/cook, until ready to eat. Originally, I tried officially 'smoking' them.. That is to say, running them through the smoker until they are done and ready to eat. It never worked out too well. I have found that the 'bugs' are much better cooked in a traditional way, such as broiled or grilled. But I add an initial little wrinkle, that makes all the difference in the world..
First, I tail the lobster, then split the tail shell to remove the tail meat. I run the 'naked' tail through either a 'cold' smoke, or a hot smoke that is maintained at a very low temp, so all I am doing is imparting smoke, not cooking the darn thing. I keep the temp down to the lowest level that will still generate smoke.
After I have run a pan or two of smoke against the tail, at as low of temperature as I can, I feel I have enhanced the tail as much as I can. For all practical purposes, it is still raw. Just nice and smoky.
I then baste it well, with garlic and butter and insert it back into it's tail shell.
Once the tail is reinserted, I use the BBQ Grill or the oven broiler to do the actual cooking.
The results is a Lobster Tail that looks traditional in every way, but has all the advantages of having been (behind the scenes) smoked for flavor, basted in butter and herbs and broiled or grilled to the final perfection, before serving.
Try this and let me know what you think. If you are disappointed, I will be very surprised.