About CL, it's a lot different story than BL. Cards are more splashable, more able to run off-mark, and shards become more powerful than in BL. That's why I think we should limit shard usage in CL, if not banned entirely. In BL, shards are okay, as they have a reasonable cost.
I think there should be a rule that says:
1. All bans that were stated by a lower-rank player MUST be accepted by the higher-rank player in either League, unless they make gameplay highly restricted or impossible.
This way there shouldn't be a problem with bans. And a secondary rule along with the first should say:
2. If the lower-rank player disagrees on the bans of the higher-rank player, the match proceeds with only the bans of the lower-rank player, otherwise unrestricted.
I have no problem with joining both leagues as long as 1 player can only claim 1 prize. The claimed prize should be the one that benefits more.
I just repeated here what the OP said.
Combining my first and second point, there is an option that if anyone wants to ban 1 or more shards (mostly in CL), be it the lower or the higher-rank player, it must be accepted no matter what. This rule would overwrite everything previously stated.
After the first month, people in BL gets their own standings based on rank. Those who are in the lower half tend to become "suddenly and mysteriously inactive". There are also the ones who are in top ranks, but avoid duelling to keep score. To avoid being a douchebag, I don't press anyone to play with me and neither do others in BL who just play for fun. But this way, against those who exploit rules and/or human emotions/feelings, I suggest the following rule:
3. For every 4-day-intervals, in which a player didn't play a single match, his rank or score drops down automatically.
The interval length and the drop in score/rank may vary based on LO opinion.
I also had an idea that rewards activity. In KotH, winners got 3 points and losers got 1. How about making the losers in Leagues get minor points too? Or would that make the gameplay much different?
A minimum-score limit should be added to CL, just like Odii said. I have seen a lot of newbies who think that upgraded beats all, therefore they either instantly upgrade or sell all their cards. There are a lot of newbies out there who have only 2-3 decks, but all upgraded. I want to exclude these kind of players from CL.
That's all for now. I hope it helped.