L;
If you go with the solar and a house battery then no, the alternator is not essential.
However, with solar it is a good idea to have a supplemental charging plan. You need a way to last through a period of low battery. Inclement weather is a common cause of low battery.
I ran that way from 2010 - 2016. I installed a charge line back to the house in 2017 but kept the OEM 105A alternator. I have always had a 120V charger and, for the recent five years, an inverter generator.
You could:
1. Compromise and eat-out until the battery is replenished.
2 Locate a source of 120V and use a simple battery charger of at least 10A. Allow 12 hours that could be spaced over a few days. Replenishment is a subject unto itself.
3. Add the alternator. This requires adequate cabling to the house battery, and we are back to that subject again. :)
With your limited roof real estate, it will be a challenge to fit solar panels. As I said before, I had two 85W 12V panels on my Safari. Not because of their Wattage, but because they fit between the roof racks to give a lower profile.
A regular 24V panel of 200-250W will be 66"x39" and will fit nicely also. I didn't go that route because of the WAY higher shipping cost plus it requires a more expensive MPPT solar controller. If you could drive to Norco(solarblvd.com), you could save on shipping. Also check local craigslist.
I strive to replenish at least weekly. In practice, when dry-camping, I will run the genny for bulk charging. Bulk charge means that the battery is at ~80% state-of-charge(SOC). The solar can complete the replenishment if the genny is run early in the day to leave adequate time. Yes, that last 5% is important! I have dry-camped for 31 consecutive days with the genny as support.
HTH;
John