If you are worried about the gap between the girt and the housing, change your joinery design from a tenon with a peg to a wedged thru dovetail tenon. And plan your wedge to be driven in from the inside. Then if the girt shrinks and the gap grows you can pound the wedge in from above the tenon to lower the girt onto the housing. Also or another method would be to draw bore the tenon with it off set more to the bottom. Then make long tapered pegs and if the gap appears pound the peg in more to lower the girt onto the shoulder. (I’m not sure if this would work.) Another method for compensating for the shrinkage is to cut a hollow in the middle at the bottom of the girt at the shoulder.
In Steve Chappell book “A timber Framer’s Workshop” starting on page 144 he talks about shrinkage of lumber and beams. And how to calculate the amount it will shrink and how to change your joinery in the designing state to compensate for the shrinkage. I reviewed it and the bottom line is you’re going to get some shrinkage no matter what you do. But you can lower the amount you see by changing your joinery. He mostly talks about how it will look not how it will work or effect the strength of the joint. If you don’t have this book you could get it and review this section and hopefully it will be helpful.
At a timber framer guild project last summer, the instructor had me hollowing out housings on tie beams where braces and queen’s posts attached to compensate for shrinkage. My point is that this hollowing out is a standard procedure to compensate for shrinkage. This barn was made of both white pine and hemlock they were fairly green timbers. And the whole frame was draw bored.
As to pegs, most of the projects I’ve worked on used dry pegs and green timbers. So the moisture in the green timbers was sucked up by the dry pegs that made them swell up and get very tight. We were told not to stop pounding them in (half way for example) and go for lunch, and come back and expect to finish pounding one in. It wouldn’t move after that amount of time. In all my reading and viewing pictures of joints and pegs I’ve never seen any pegs lose or falling out of holes. Most pegs are cut with a drawknife and have edges to grip the hole; this was to help prevent the peg from rolling out if the building racked in the wind. Although I have seen and used turned pegs but those were the dry ones, mentioned.Good luck