That is, keep it as it is. Thoose that are good/best, will eventually be in first league. It might take some time, but hey, that's life. If they really are good it won't take too long a time (perhaps in Go 19x19), so where's the problem?
The way it is now, everybody can play to move up as soon as the previous tornament ends. With what you're proposing, a few players can play to move up while everybody elser has to wait around.
So it doesn't actually do much of the players you're trying to seed, but it's a royal pain for both the players already in League 1 and the players who aren't highly rated.
This is good idea for go19x19 (one round ~ 8 month) and there is 4 leagues.
For other games is possible combination of monthly cup with chamionship (winner promote to 1. or 2. league).
The same structure is in (ICCF) where winner of World Cup promote to 1/2 final of World Championship, or winner of European Championship promote to 1/2 final of World Championship.
How do you want to make the combination of championship and monthly cup precisely?
Let me assume that one round in a championship lasts 4 months, so during 4 months you get approximately 4 winners (sometimes more and sometimes less, it depends) of monthly cups. Then all winers should promote to 1st league? If yes, then you have in 1st league approximately 13 (4+9) players...
This is not main problem (in go 19x19 1. league play 13 players now). Main problem is, that for several strong players is way to first league very long. ICCF solve this problem with combination of tournaments and with tournament, where can play players with some rating (Master Norm tournaments).
In my opinion, the rule that the winner of monthly cup promotes to 1st league doesn't solve this problem (too long way of strong players to 1st league) at all. Already now the monthly cup has 2-3 rounds. Thus to win the monthly cup, which has 3 rounds takes exactly so much time as to get from the 4th league to 1st one.
I guess providing a shortcut to the top leagues in Go 19x19 makes sense, since it would take more than two years for a new, strong player to promote to the first league. But in other games, I don't see the point.
In Hex, for example, there are currently three leagues, and it has taken roughly three months from the start of one championship to the start of the next, which means it takes less than a year to promote to the first league for a new player. That is acceptable, I think. (Besides, the suggested change won't have any effect at all, since David is going to win all the monthly cups anyway.)