A car can be insured whilst SORN'd. You do need to get an MOT to get tax after all so there is no legal reason why you cannot claim on the insurance. How much is your excess and are you protected NCB?

If you do claim, then you WILL need to keep the old tyres to prove condition, and if they are getting down (below 5mm normally), you would be expected to contribute towards the cost of replacement.

However, if it was not insured, down to you.

I agree though, I would not let the garage sell you the tyres, although they would be within their rights to charge you storage after 24 hours have passed since the MOT being done, so you might want to be quick about it.