We disburse all of our insurance premiums for the month on the 1st and haven't had any issues. We have a couple of agents that have asked that we sent in the money before the due date so we pay those manually so even though the policy due date is Sept 1st, the escrow analysis will show it paid in Aug of each year.
As for the taxes, it works the same way, the analysis will show the disbursement date and future disbursements will fall on that date.
The only thing that I can see happening is they may have a shortage the first year because you disbursed earlier than disclosed but it will work itself out by the next analysis.