Apologies about the novel; I spent a lot of time thinking about this, and the more I wrote, the more I thought about it. Also, I ask that as you read this, please assume that I am intending this with all the best intentions! If I come across negatively, please know I'm simply being philosophical!
For me, this answer has to address another more fundamental question:
Do you have too many characters? Since you need a method, that means it isn't something that would occur inherently. Go through each of your characters and decide if they are absolutely necessary. Maybe some could be merged. Maybe some could be cut all together. Maybe some characters make more sense as NPC's that follow the party around from hub to hub rather than playable characters. You get the same story value, but eliminate any anxiety about the player limiting to only a single group. Maybe you've got 10 really cool class ideas, but only 7 important characters: Save 3 of those class ideas for the next game. If the character isn't necessary, get rid of it!
Assessment complete?
Then let's move forward and assume each character is important, or assume that even if they aren't all unique snowflakes with importance, something about the game is inherently more fun by having a larger cast of playable characters. Now we need to answer another question:
Why would I want the party members to switch? (I don't mean this as a rhetorical question).
The answer to this can't be: I've made 100 characters and I'd like there to be variety because here's the deal:
We make games (hopefully) because we want others to enjoy them. If somebody enjoys playing a single party through the whole thing, that's their right and luxury. If there are characters that aren't intrinsically important to the story or the fun factor, let the player do as they wish. If the game is good, you might get new playthroughs with different "super groups." Do you need switching?
THAT SAID
You've thought about the above, you've made the changes you want to make (if any), and you definitely still want to encourage switching. Here are my thoughts:
1) I like the idea of battle-swapping. To me, this feels like everybody is in your party at all times, so it isn't really a swap. Then you build all mechanics around the assumption of the entire player pool.
2) I dislike the make everybody level up together unless you are coupling it with a railroad technique (such as Hard Counters -Frost mage needed to fight fire baddies,etc-, Required Characters for Specific Scenes, etc). If you are going to use a system like that which forces a player to swap (or the game will suck), then you should make it as painless as possible.
3) I'd encourage creating a mechanic that makes character swapping appealing to the player. Examples:
A) Let's say the player is commanding an army. Every character that hits level 10 becomes an officer. The more officers, the better the gear held by the shops. Therefore it becomes in the best interest for the player to try leveling everybody up... Doesn't have to, but he's missing out on something.

Another example might be the "win" mechanic comes by having Overall Party Level hit 500. Since levels get further apart, the player has interest in trying to level horizontally rather than vertically.
C) If a character passes a certain level, they "graduate" and become no longer playable on this arc/campaign/game. This makes focus on a single character dangerous.
4) Split your parties up and have them handle different parts of the arc at the same time. Then your party isn't exactly swapping: You are simply pursuing multiple paths simultaneously.
Just some thoughts... Good luck with it!