I guess as electronic tech I think outside the box sometimes. How about a higher capacity battery? Or, an isolation diode and second battery just for the cab & lights/fan?
That battery could also be trickle charged when tractor is parked.
Before I did anything though I would disconnect battery, either + or --, then series an ammeter. Jumper ammeter to start motor
(High current jumper!!!) then remove jumper.
Better yet simply an inductive DC meter like
Fluke 362. Then try various things...lights with fan, etc. Compare that to alternator output.
You could actually get an idea just with a voltmeter across battery, not as good as ammeter, but say it's 12.6v...start tractor & now it's 14.2v...turn everything on and it's 12.9v you know battery is still being charged.
If tractor is going to be used a LOT running fan, lights, etc. maybe then I'd consider a higher output alternator.
I like going from simple to complicated since as others said wiring should be upgraded also.