Heian Dai Shogi really is not a very fun game. And I truthfully doubt I would ever have any interest in playing any of the very large games. I'm surprised that you wouldn't want to tackle Taikyoku Shogi, though, as that is far larger than Tai. From what I read, Dai isn't too good either, as Chu Shogi has almost all the same pieces, but is on a smaller board and thus is more dynamic and quick paced (not just fast, but I mean action-packed instead of dull). Tenjiku is intersting to me, as it has some very strange and unique pieces not found elsewhere, and Dai-Dai is supposed to be pretty good even though it's even larger than Dai. I might be willing to try those two, but I'd probably not want to get any larger. You've probably noticed, I have much less time lately, and I've been playing poorly because I've been giving my games much less attention when I do move, so it really could last a very long time playing Dai Dai (I hear Tenjiku is actually really quick). Anyway, we'll see how it goes. Good game, btw. You got me!
I would still like to try some of the smallest games we never got around to as well, like Tori, Dobutsu, Nana, and Gufuu shogi. Like you said, we could alternate. You decide what we play first.