While you most certainly can do it at 300-350, you will not get much smoke flavor... Not that it is a bad thing, grilled or baked steelhead is great! But if you are wanting a smoke flavor but not "smoked", then drop down to 250 and use a rich smoke like hickory. I have done both large fillets and whole rainbow that way, it is pretty good and a great way to use the smoker in a pinch. You do not need a brine if you are going to do it fast. Now realize it won't take long at all! For a typical sized fillet, maybe 20-30 minutes? Possibly an hour if the fish is thick and cold. At 350 it will take about 15 minutes for standard "medium rare" salmon (steelhead is in the salmon family but is really a big water rainbow trout). If you like your fish fully cooked and flaky then maybe 20-25 minutes.
Personally, I like to brine 2-6 hours in a basic salt/brown sugar brine and then smoke at 160-180 with 50/50 apple/hickory. Takes about 3 hours for whole small rainbows, or 1.5 hours for fillets. I actually have some wild caught whole rainbows (gutted of course) brining right now that I will rinse shortly and get ready for the smoker in the morning. After they are cooked I peel the skin and pull the meat off the bones. With a little dill weed and some cracked black pepper it makes a fantastic snack with some good crackers!
If you do it at 250+ be sure to season the fish well, as you are pretty much just baking it in a smoker. You will need some seasoning to help impart some flavor that will be lost by not smoking for a longer time.
Good luck and enjoy your trout!