Yes, high-output alternators can be installed on your F53/Triton V10, but you'll need to upgrade wiring and possibly the serpentine belt system. The Tucson unit you found is a solid choice for your large inverter battery bank needs.
Intermittent Greyhawk AC with working fan indicates compressor clutch cycling. Check refrigerant level, clutch engagement, and voltage at the clutch coil. Low refrigerant is the most common cause.
Your East Penn 6V AGM batteries likely sustained some damage from the complete discharge to 8V, but your charging behavior suggests they may have recovered reasonably well. The key indicators are how they hold voltage under load and their actual capacity.