Slightly buckled wheel

[timg147]

Hi all
Looking for a bit of advice. I own an audi s6 avant (full fat v8 turbo version) running 20" wheels.
At the mot I had two advisories both relating to the same wheel. Firstly low tyre tread, on closer inspection they are all getting low and being quattro will need changing together. Secondly a slightly buckled wheel, I have stuck my head under the car with a torch a few times and also moved the car backwards and forwards and can't see anything obvious. The mot centre said the tyre place down the road could knock/tap it out. I have used the tyre place a couple of times for the missus horseboxes and they have been really helpful.
Just wondering if it's safe to have them tap the buckle out?
Am I best to get the tyres changed first or the buckle sorted as the wheel will need balancing after. I will give them chance to quote for the tyres but not sure how competitive they will be as I'm looking at michelin cross climate 3 sports or goodyear asymmetric 5's.

[Landlubber]

I would have thought stick a spare on, get the wheel fixed then get a price for tyres.

[timg147]

I might be wrong but I think the spare is a space saver 😕

[E-bmw]

I might be wrong but I think they can still straighten your wheel out if you have a space saver fitted on the car.

[ARH]

There are plenty of wheel refurb places around that will fix the wheel properly, a tyre shop will not be dealing with a buckled wheel correctly. Its an expensive fast car why not just get it fixed properly?