John, your omnivent should work without a battery, it's probably time to do some fault finding, start with your fuses. Taking a battery with you is a good idea in case of unforseen problems with the site hookup. The caravan relies on 12v quite heavily, so not having a standby is a bad idea. Also, a battery helps smooth out the output of your vans 12v supply.
Modern caravans are fitted with a 13.8v power supply. Caravan manufacturers call them chargers but they'll only charge a battery to about 80% of capacity.
The advantage of a power supply is that you DON'T need to have a battery in place although all the outlets selling batteries will tell you otherwise.
Omitting a battery will save weight and cost BUT makes you totally dependent on the mains hook-up, not always a good idea if some other caravanner has tripped the site mains out! It also means that you've no power if you stop en-route.
If you ALWAYS use a hook-up, I'd suggest just fitting a small 36Ah car battery - they're cheap, lighter and will provide a bit of power when the mains fails. If it's not discharged like a leisure battery when off hook-up, the car type batteries work fine - I get 10 years life out of mine!
So, just to check, can i assume from what's been said that it's OK in principle to hook up to the mains with no battery connected in the van. Everything else 240v will work, but presumably the 12v stuff will be dormant??
Depending on how your caravan is wired the 12v systems may still work without a battery connected. In most 'vans the battery is only there to stabilise the voltage, and provide a reserve. Some 'vans are starting to switch to mains lighting now which rather obviates the need for a battery other than the water pump. My present 'van has no provision for connecting a battery and I've never given it a second thought.