Lighting:
12V LED lighting running on battery is much more convenient than having to start the genny just to see what you are doing. If the desired fittings cannot be obtained in 12V form, a small inverter powering 230V lights is not impossible, although the power consumption may be significantly higher than 12V direct
Cooking:
It's not really practical to operate heating / cooking equipment via an inverter from a battery unless the battery is very large indeed. A normal 100-120Ah 12V leisure battery cycled moderately i.e. not pushed to the limit, will run a 500W appliance for about an hour. Running a 1kW appliance even for half an hour will stress it and shorten its life and as it ages you won't get half an hour anyway. Therefore you must plan for the full capacity of your cooking equipment to be handled by the genny. Thermostatically-controlled cooking will cycle on and off and if there were many appliances running on a big genny you would work with the average load after diversity. But with just two units you can't make any economy because the genny will have to be large enough to power both heating elements at once because this will occur frequently.
Water Heating:
Likewise you can't realistically heat water from battery, A leisure battery contains enough energy to heat 8-10 litres of water for washing up, if spread over many hours to give the battery an easy life. However from your wording 'hot tap' I think you mean to near boiling e.g. with a Quooker-type dispenser, in which case you are looking more like 6-8 litres max. With stored hot water you could arrange a priority relay that won't allow the water to heat while the grills are on, to avoid having to actively manage the power switches yourself or have repeated trips or brownouts from the generator if you overlook the power management when busy, Also with instant if you are prepared, as you suggest, not to be using water when grilling.
Refrigeration:
I can't honestly recommend leisure type 3-way (gas/12V/230V) fridges, which use absorption chilling rather than compressed refrigerant. They are not usually intended for commercial use and have a limited cooling capacity, If frequently opened or loaded with warmer produce and especially drinks, they might not maintain temperature. For commercial use excursions about 5 deg.C are not permissible and will give you a headache with maintaining your temperature log book. Commercial absorption units of much higher capacity are available but they are specialised and expensive. On 12V the consumption of even a small fridge is in the order of 8-10A and can only sensibly be used to maintain temp in transit, with the vehicle engine running. Otherwise it's 6 hours off a leisure batt max.
12V compressor fridges can offer low consumption and good cooling performance and we used them for commercial catering. They are expensive compared to equivalent 230V fridges. Typical running load for a 100 litre fridge would be 2-3A, so quite practical to maintain temp and run from a leisure battery even if the compressor runs continuously. Even a good new 230V fridge running on an inverter can be satisfactory, although the inverter (not the battery) must be sized for maybe 500-750W to handle the compressor starting load. On my boat I have a 12V fridge and a 230V freezer. The fridge runs 24/7 on battery, I don't have to think about it. It got left on once while the boat was moored up and was still running 3 weeks later (I have 600Ah domestic battery). The freezer is switched off when not in use but runs happily on the inverter.
Options:
My suggestion would be to use 12V for lighting and refrigeration on the move (via an inverter, with mains fridges). Then transfer the refrigeration to the genny when running and leave the lighting on 12V powered by the charger/rectifier also on the genny. If the genny is big enough for your grills, it won't be fazed by starting the fridge compressors.
Water heating is the fly in the ointment. What I have fitted on the boat is a variable-load stored system where I can select 500W, 1, 2 or 3kW input according to how much power I can spare for that purpose. I can heat the tank quickly on a 16A shoreline, then switch to 500W max to keep it hot and replenish water used for hand washing and washing-up. The temperature is not affected, only the rate of re-heating. It has the advantage of not being the only heat source - I can heat it with the LPG central heating boiler (expensive) or with the engine waste heat (free when cruising) so the electrical heating is only normally used on a mooring.
For compact, portable generators, it is possible to get asynchronous units that can operate in parallel, E.g. at work we have a number of Honda Eu20i 2kW units that can be operated up to 3 in parallel, for 6kW total. Obviously the purchase cost is higher than a single unit but they are super gennies, very quiet and reliable as far as petrol units are concerned. And of course you can pick them up single-handed, which you can't do with a 6kW diesel. The diesel will give better long-term value and useful life though.
One trick that some motorhomes do is to carry a heavy diesel genny slung under the body on a winch. On arrival the winch is lowered and the genny wheeled away. A few fair organ owners park their genny on-board the organ and wind it up and down a tailgate ramp with a car trailer-style winch.
Enough ideas for now!
