Change of insurance?

Moin! At the moment I have the following situation: We have a car, which is admitted to my wife as a holder, however, insured in my name. Now I want to get a vehicle with truck approval soon. VW Caddy, Seat Inca, Fiat Scudo or something like that in the direction. This vehicle must then also be insured as a truck or van. Incidentally, the vehicle is not used commercially, but exclusively private. Now to my question. And now it’s complicated. At least for me.. I found an insurance that insurances vans at a good price. Can I change the insurance now so easily, because the current ones do not insurance my vehicle or how does it work? I wanted to take the car with me to the new insurance and run as a second car. By the way, the car on my wife falls flat. Too expensive because SF-Class M I hope you can follow me.