Hello Chris,
Bearing in mind that you say you have LED lights and don’t use a television, I am a little surprised that your battery only lasts about 4 days.
Traditionally although conventional lights are typically only each rated at about 1A (12W) they tend to be used for long periods of time, so they can be a substantial drain on a battery, but swapping to LED’s you are likely to only be using about 0.1A or a tenth of a conventional bulb or tube.
Surprisingly the items that use a lot of current (e.g. water pumps) often aren’t used for long, so the they don’t use very much battery capacity.
However there are other items in most caravans that can and will use modest amounts of power over long periods, and it may be worth looking at their usage
The fan on the heating system will use about 1.5A when running at full speed,
Less so than the heating fan, your gas powered water heater will also use some power, so perhaps only turning it on half an hour before you need the hot water, and off again afterwards will save some power.
In some cases the radio can use modest amounts of power even when apparently turned off. This is used to keep the station memory active. A portable radio with its own batteries may be an alternative solution and you could then completely disconnect the internal radio.
The other problem I can foresee is the way you are trying to recharge your caravan’s internal battery. The system you are using has several in efficiencies, which means your caravan battery is unlikely to be properly recharged. For various inefficiencies in inverters, chargers, and the ability of a battery to retain power used to charge it, you will be doing very well if you get anything close to 40% of the cars battery energy transferred to the caravan battery.
With the above in mind, your systems is not giving you the best performance.
May I suggest that rather than trying to recharge the caravan battery through the inverter, actually swap the batteries between car and caravan, perhaps every two days.
Parkseys suggestion of solar panels may also be an alternative – and silent. You would need at least a 60W panel.
As for generators, ask your local dealer to demonstrate. If they want a sale they should oblige, but don’t expect silence, they do produce noise.