Free: Contests & Raffles.
You'll want to be careful because there is no fat like the typical meat that you smoke. Brisket, ribs, pork butt, etc. all have a lot of fat and are smoked until well done.
Haventbdone backstrap, but have done tenderloin. I just did salt and pepper, smoked with hickory at 155* until the internal temp was 130*. That takes about 3 hours. I then did a quick pan search (only because I rant out of propane). Turns out fantastic. You get an even cook throughout and it picks up the smoke flavor really well. I wouldn't want to do a real long smoke though.
Only way I do it now leave blackstrap whole wrap in bacon smoke on Traeger for an hour then turn up heat until internal temp is 129 can paint up with bbq sauce the last 10 minutes if desired
If you want the smoke flavor maybe you can do a short smoke and then cook that backstrap on the grill. Not sure I would want to risk drying out the best cut of meat that you will get off an elk by smoking it.
I always take some deer backstrap and cut into 3/4" steaks, lightly season and smoke for about an hour with apple chips. Then I put it in jars with an onion slice, cover with au jus and can it in the pressure cooker. This meat is always saved for elk camp and it simply heated by removing the lid and placing the jar on the wood stove until heated. A pint jar per hunter makes for a quick and easy lunch or supper if we get back late. I'm a pretty popular guy when I break it out every year.