Even though the voltage shows 12v + through a multimeter you will find that as soon as a load is applied then this may drop to unacceptable levels. This is probably due to the extra resistance and possible connections all the way to the rear plug. When we did a month in the Beach with a Waeco/Mobicool FR-35 31L fridge I found that the same issues using the 12v cig plug type fitting***
It is *much* better to simply dismantle the 12v plug on the Waeco, solder on a decent length of twin core automotive cable (high amp rating to avoid drop) and take this directly via an inline fuse to the leisure battery. In this way you avoid all the inter connections that add up to lose current.
https://vwcaliforniaclub.com/threads/fridge-for-beach.13960/page-3#post-183713
***On our pod I wired a supply to the water pump from the B pillar 12v socket and then an extension from this was intended for the fridge supply plug but it was one connection too many in terms of V drop:
https://vwcaliforniaclub.com/threads/building-a-kitchen-pod-for-the-beach.14752/page-3#post-160530
I have to say that I cannot see why you could possibly want a gas powered fridge when the modern 12v/240 compressor types are so clean, easy and efficient.
Ours was great for four people and like others I turned it off at sleep time and back on in the morning.
Check this topic below and maybe think again on the gas 3 way thing - it's old tech and as others have said you don't want to run it inside really.
The 12v compressor types are not that power hungry - a drive now and again or a hook up and charge from a Ctek once a week or two is all we did.
https://www.amazon.co.uk/dp/B00FC42HAA/?tag=eliteelect-21
https://www.amazon.co.uk/dp/B001XPHO5A/?tag=eliteelect-21
Using the above hook up and battery charger (infinitely better tech than any in-built VW Cali battery charger) is a really neat and compact and cheap 240v solution. The Cteks come with plugs/adaptors you attach and leave on the leisure battery - no fiddling with crocodile clips, just leave the rubber plug under the passenger seat battery rubber cover and plug in as needed. Always carry a multimeter too - very useful to keep proper tabs on voltage/usage etc.
Good Beach fridge advice here too:
https://vwcaliforniaclub.com/threads/fridge-for-beach.13960/