I wouldn't add it to the priming solution. I usually add the priming solution to the carboy, so it would be hard to split it.
For the tea, I'd steep it long enough to pasteurize it (20 mins @160?) so you don't add any nasties.
Heck, you could make a tiny tea and add a wee bit to a commercial brew to see if you like it. It won't be exactly the same, but it may help you decide before you make 5 gallons that you won't like.
In my experience, both oak and peat take a while to mellow, so if you drink it after only 3 weeks, it may be harsh. Stash a few and try them in 6 months.
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