Can I run 12V camper van fridge whilst 'trickle charging' the battery in van?
Make sure that when you switch over, the battery is still in the circuit. Check that.

You can't run the fridge with that battery and that trickle charger. It would work if you had a much larger battery, or a much larger charger, or much smaller fridge, but not with the equipment you have.  When the fridge starts to turn on, it draws so much current that battery voltage drops, then when it turns off, the battery voltage swings up.

Having said that, here's something you can try: put another big battery on the trickle charger, and leave it there.  When the fridge tries to start up, hopefully the big battery will stabalise the voltage enough. I'd also add a circuit breaker and a big coil of wire, to handle the surge when the two batteries are switched in parallel.