Hi Greg,
First, I wouldn't use a brine. I find them of no benefit for cooking fowl.
Second, I would recommend hot smoking rather than cold smoking. If you haven't already, click into "According to Smoky" and read the "Glossary" for the distinctions.
Cold smoking is more for preservation and when I cook a pheasant, I expect to eat it shortly.
I actually prefer roasting pheasant. The bird is read to consume in about 35-40 minutes. It is always a welcomed treat and that is long enough to wait.
Remember that the pheasant has less fat interspersed in its muscles than a commercially raised chicken, so basting with a little oil is beneficial. I like to stuff a small apple and a small onion, quartered, into the cavity along with a stalk of celery. Then rub a little oil on and salt and pepper the bird. I roast at 350° until it reaches 160° in the thickest part of the thigh then take it off and let it sit for 8-10 minutes before carving.
If you want to get fancy, sprinkle on a little fresh rosemary or thyme.
Be certain to remove all the pin feathers and carefully clean any shot paths.
Have fun,
Smoky