First time I came across the diesel cab heater was 1980, seem to remember a Hunter make, fitted as standard to a Grove crane, mainly as too hard to get cooling water to the cab. The Eberspacher seemed to be the industrial standard, but Webasto is a close second, however there must be hundreds of models each slightly different, heating water or air, some with more or less features as others, and some do have an under voltage cut out, and also different methods to atomise the fuel and ignite it.
Also, there are many different types of lead acid battery, roughly we have the traction battery, and the engine start battery, the traction is designed to deep cycle, and low amps, and engine start not deep cycle that can damage it, but high amps, and the leisure battery was a halfway between the two. There are also gel batteries, often used for the heaters on radio valves, and the valve regulated lead acid (VRLA) similar to the absorbed glass mat (AGM) which were originally used like the traction battery, alarm back-up, emergency lights, mobility scooters etc. But are now used with cars using the stop/start technology. The AGM does seem better with deep cycle to the flooded car battery, but one can't group all lead acid together and make comments like over discharge will damage them, as it does depend on the type, however, lead acid in general is damaged if left in a discharged state.
Also, the charge rate of lead acid is slow, even with flooded traction batteries where slight over charging does not matter, we look at 8 to 12 hours to recharge, and it depends on how long they have been left discharged, I found a pair of VRLA used with a stair lift took around 10 days to charge, and was monitoring with an energy meter, and was as if someone had flicked a switch, 9 days nothing, then charged on day 10 and seemed as good as new, but in the past I would have scrapped the battery before that time had elapsed.
Back in the 60's and 70's before the car alternator had hit the scene, we would give batteries an equalising charge around every two months, we would use a trickle charger overnight to ensure battery fully charged on all cells, and top up the cells, this also helped stop the sulphur going hard on the plates, and this is the problem when using a charging system which may not fully recharge the battery every day.
If I look at my solar panels, every night they are recharged using off-peak, but during the day, solar this time of year sometimes fully recharges them

but

some days the solar only slows the discharge rate, so I could go a week or more without fully recharging the batteries if using solar only. With lithium not a problem, but this is a problem with lead acid. Using a solar panel to counter self discharge with a caravan battery in winter may work well, but to put enough in it for daily use, is really not going to happen.
So the battery will likely need charging from a mains supply from time to time, so looking at what size battery can be transported with ease, I have three sizes here, 12 Ah, 35 Ah, and 90 Ah I can carry them all, but would not want to carry the 90 Ah far, the 35 Ah scooter batteries are about the largest I would want to move regularly, but since the 12 Ah will start a petrol car, it would also fire up a heater, and as a battery pack, easy to carry.
If we look at solar panels, 100 watt = 650 watt/hours height of summer on a bright day, about 55 Ah so having a battery over 35 Ah it is unlikely to be fully charged even in summer, winter again on a good day 9.6 Ah so even the 12 Ah battery is more than it could recharge in a day, in fact a lot less, as just not enough time, so using solar and lead acid 100 watt panel and a 10 Ah battery would allow lights to be used but would not run a heater for a reasonable time, you will need to charge battery from mains at some point, using lithium which can be charged faster you have a chance.
The Tan-y-Llyn railway use diesel heaters for their coaches, the battery does not last out the day, it needs charging each night, our railway uses steam to heat carriages, and should we have to diesel haul then no carriage heating, so we have looked at diesel heaters for the Santa specials, we have the annual get together on 29th this month, where all the North Wales heritage railways join for a day of drinking and merriment on the train, this year at Llanfair, so I will ask the guys from Tan-y-Llyn what size battery they use. In the main, steam is better, as it gives some ambiance, but diesel would be a good back-up. We have larger carriages to the others, as no cliff faces, or bridges to restrict the width, the new to us observation carriage, will likely need some heating, so it needs thinking about.