Hi,
Seems to me you have a good size lawn tractor!

The alternator will be rated to the total load capacity of the electrical system with some reserve as to not overheat the alternator.
The PTO will draw the most amperage continuously and a slight amount for the ignition.
Battery charging and headlights are additional.
The starter motor and horn are considered intermittent loadings and are of no consequence.
A battery is like a shock absorber in an electrical system to work with surges of power requirements.

You need to at least meet the PTO load, of which, will probably be over the nine amperes.
The total load should be at or over the nine in my estimations.
If it had a 13 alternator the load is going to be less than the 13 provided.
IMO I would not change it without a ampere reading and considering removing the headlights wiring altogether as a must if you go smaller.

As far as charging the battery before each use, it should be kept on a maintenance charge all the time.
If not, you will buy more batteries during the lifetime of mower.

Hope this helps with options.

Phil