How would kicking off End of Month processing prematurely (and during the day) affect the day to day usage of the application?
If you start End of Month processing too soon, the data you get will not be accurate, as it does not represent a full month's worth of data. As for doing day to day work in the application, it is likely to slow the system down as long as the process is running.
Most importantly, however, you must immediately notify all users in the system to no post any transactions while this is happening, especially Batches. This will put a lock on the tables that the EOM processing is using to gather data and the whole system could potentially freeze up. Users will have to wait until they get the "all clear" from DMS that EOM is completed before they post any transactions.