I bought the same sprayer this spring.
4.5a is not a lot of power. About 54 watts. Less than the headlights, so unless you are spraying at night with the lights on, your tractor will likely handle it just fine. Almost every CUT I looked at had a 30-50 amp alternator, and all the cooling, fuel pumps, etc are mechanically driven. The tractor is not drawing a lot of power on a regular basis besides a few relays, the cruise control, and brake lights so it should be able to keep up with 5 amps just fine.
Make sure the "always on +12v" is wired on the 7 way. It's not necessarily required and on my lot of trucks, you have to manually add a fuse if you want to use it. Not sure if they automatically wire it on a tractor 7 way. Make sure the fuse and wire is rated to at least 7.5a so you don't melt the wires if you are running it for hours.
FYI, most deep cycle batteries are between 60 and a 100 amps depending on group size. The RC number is the number of minutes the battery can sustain about a 20 amp*hr load and keep the voltage above 10.5 amps. Your pump probably isn't rated to provide the right psi and flow at only 10.5 volts and there aren't any ratings for run time at 5 amps and over 12 volts. Bilge pumps I looked at to move water into tanks had noticeably different head and flow ratings for being power from a running engine vs just battery voltage.
My advise is to make sure you keep the switch inline handy because the only way to start and stop the sprayer is by cutting power or getting off and turning the valves.
I ran mine the first time off the rear work light prewire and it did fine. But because I wanted to run another pump that that needed more than the work light wiring can safely provide, I just ran a new line to the battery with a fuse right after the battery connection. I would suggest using the tractor power instead of a free standing battery unless it is your only option. The sprayer pump will be much happier using the alternators 13-14v instead of the 12.2 to 12.6v of a free standing battery. At the higher voltage, the current draw will be at a lower amp rate and will be able to maintain pressure and volume better and will run cooler.