It will all depend upon the current draw of the sprayer motor and the output of your alternator.
Worst case is that over time the sprayer will prevent the battery from charging fully while you mow.
I can not see a problem.
Got a couple of lawn care professionals with 5 gallon spray unit mounted on a frame over their engines so they can spray while they are mowing.
Does not seem to cause them any grief.
Pump connected directly to the battery with a switch.