There will always be tears no matter what you do. Two or three seasons is a short timespan, the game is supposed to be played in the long term.
From my observation, there are game mechanisms already that will 'correct' an unbalanced league. Swapping 3 or 4 teams in a conference is one and completely redraws the equilibrium.
Let's see your own league, here is a breakdown of past winners :
s12 : big8
s11, s10 : great8
s9 : big 8
s8 : great8
s7 : big8
s6 : great8
s5 : big8
s4, s3 : great8
s2 : big8
It seems remarkably even to me. You know the saying, if it works, don't fix it !
About playing the same teams over, on the contrary, because of the leagues structure, there is a high turnover of teams.
Let's look at your league again. You were promoted on s11, let's see how many teams are still there since s10 : only 2 in big8 and 3 in great8.
Things change, the game is not frozen only a bit slow paced.
Sleet, don't be so defeatist, cheer up ;)
Your big8 had a champion on season 10, not that long ago.
If you're doing a better job than other managers to improve your team, there is no reason you won't beat them whatever conference you're in. Actually being in a relatively weak conference will help you to get HCA and maybe enthusiasm build up.
Usually, in leagues where the winner is always from the same conference what happens is that the demoted teams keep winning the league over and over again. And of course, because the promoted team is swapped for a demoted team, the winner always ends up in the same conference. It's not a problem with league balance, the demoted team would certainly be champion in the other conference anyway. It's a sign that the league is somewhat weak and that there are no team competitive enough for the higher level.
I see no problem here and moving teams around seems too complex and needs good reasons for it imho. Conference dominance can shift quickly from my experience. Just my opinion.
You can be credited for an articulate suggestion with a workable solution though :)