I would try modest amounts through multiple accounts, e.g., 5 checking accounts absorbing $4k per week each in MOs. You could also seed the deposits with checks drawn on the other banks, but be careful not to kite or raise anti-kiting filters.
Also, if you have the liquidity, consider buying short-term CDs (3-month) with the funds accumulating in the checking accounts, rather than immediately using to pay CC bills.