This will be my first tTBH too this year. I elected to make my tTBH just like you're describing for the same reasons. I like your idea of a strip around the outside to close some of the gaps, I might "borrow" that idea if that's alright with you. I used 2x10's as well and built it 46 1/2" long (outside dimension so a 4' cover would overhang a little at each end.) I cut two 2x10's about 1/8" narrower than the width of the TBH for follower boards. Then I attached a 1/2"x1 1/2"x 21 3/8" strip to the top of the follower boards to "hang" it like a frame in the box. (BTW I cut all my top bars 1 1/4" wide and I ripped a bunch of 1/4" plywood strips to use as spacers if/when needed.)
What I'm gathering from our mentors here is we should reduce the hive down to about 10-12 bars initially to get the bees started, then expand as needed (maybe some of the folks with more experience will chime in on this one.) I made entrances at both ends and plan on starting two colonies in each end of my tTBH with 2 follower boards separating the colonies. I'll be setting up the hive close enough to our barn, that I could run an extension cord to it and put a small light as a heat source between the follower boards during the to keep the girls warm when these sub-zero cold snaps hit. Just a thought now, I have to research that idea better.
Hope my rambling helps!