(This is not the first thread on a similar topic on here, last time it was someone who wanted to put a row of seats IN!)
Here is the Gov.uk page that tells you how to change your V5c
https://www.gov.uk/change-vehicle-details-registration-certificate
And here is the list of things you should read,
before contacting the DVLA
https://www.gov.uk/contact-the-dvla/y/vehicle-registration-and-v5c-certificates-log-books
Most minor things don't need evidence or inspection. It will take, in normal times, 2 to 4 weeks to get the new V5c back, but they may well be slower at the mo due to Covid19. Or you could try ringing them, they may be willing to change it for you and simply send you a new one, they did this for me, IIRC, when I changed a Mini engine from 998 to 1275, but that was a few years ago and long before computerisation. But you may well be on hold for a long time, only to be told, "read the website, change the form and send it in!
If the change is going to be temporary and one day you will change it back, maybe to sell it or summat, then you may want to consider what you do. It may not be a problem, you could just change the V5c back or you could just leave everything as it is, take the seats out and get on with it. But your talk of insurance makes me think you are looking at it as a permanent change.
Having run a kit car club for 15 years and dealt with registrations of all sorts of vehicles, then SVA and finally IVA, it is not totally obvious.
For instance converting to a camper is not a thing to take lightly!
Personally I wouldn't muck about, insurance companies can cut up rough and use any excuse they can point to in the paperwork. Properly
registered and properly MOTd, the insurance company will have nowhere to go in the case of a claim.
Best of luck! "Simples" it is not! (Unless maybe you are a Meerkat!)