I would look for another battery retailer, sounds like he only sells standard starter batteries? There are good lead -acid based AGM batteries on the market that are more suitable than your standard starter battery. With a suitable type battery, and enough Ah capacity there is no problem, especially when supported by solar. Been done for years by our European camper friends. Our batteries (two Varta LA95's AGM) are two years old and not showing any signs of deterioration, yet apart form charging the ebike batteries, we also use the Nespresso (1400W ) and sometimes a hairdryer through the inverter. We do have 300 W solar to support house battery charging, and try to use house batteries wisely. Making 4 cups of coffee and then recharging two ebike batteries would be hard on the house batteries and likely reduce its life.
Our Ebikes have a range of 120 km, with a 418 Wh battery (42V, 10Ah). At 75% discharge, it pulls 18A from the house battery for a short time, then drops to around 12 A and progressively lower after that. Even when charging without support from the car generator or solar panels, I haven't seen the battery voltage drop below 12.1- 12.2 V. I would guess that most ebike batteries have a lower capacity, I hear ranges of 40-60 km stated frequently, suggesting perhaps 200Wh batteries? But yours is at the higher end at 500 Wh.
Recharging our Ebike 400 Wh battery from around 70 % discharge will take about 30 Ah total over three hours. That is a bit more than desirable for most single 95 Ah batteries, the recommendation is for 20-25 Ah before charging the house battery again. Frequent discharge of the house battery in excess of 25 Ah will shorten its life (reduce the number of discharge cycles left).
Your E bike battery of 13.8ah 36v, 500 watt should be no problem if you have 2 suitable batteries around 200 Ah total. However you may need to recharge them (solar, or driving) in order to be able to use other appliances (lights/pumps/fridge/tv) without long term affecting battery life significantly.
The German RV "Reise Mobil" magazine (September 9, 2009) has doen a comparative test of 7 different 95Ah battery brands that are often used/recommended for campers. They used a test cycle of 45 mins discharge at 10A, 15 mins of 80 A (!), 1.5 hr of discharge at 20 A before recharging. Interesting reading.