Matching Tyres Does having different or un matching tyres on the back wheels cause it to make loud rubbing and flapping noises when you drive your car?

Are you measuring this to to the tyre, the rim or the actual hub? Check the silly simple stuff first. Are they matching rims? Are they matching tyres? Are they BOTH at the same correct air pressure for where you are driving, (Hard for road, soft for serious off road)? Same 'size' rims can be a slightly different shape between manufacturers. Are you using 'spacers' between the hub and rim? If you are check they are in place, fitted correctly and the same thickness both sides. Tyres, especially the bigger 'off road' ones, can bulge if they are slightly under inflated or you are not parked on flat level ground when you measured it.
If all of these are all OK you will need to get the lift kit checked as something might not be fitted or lined up correctly.

I have just skimmed a road test report that claimed the car is exceptionally quiet on the road...

By now the tyres might have been changed from the original specification - some tyres are very noisy and assuming the wheel bearings, etc. are ok the tyres are the most likely culprit. SUV tyres are expensive and good ones are very expensive so maybe the tyres are now not such high quality or are off-road or combination tyres rather than road tyres...
That is something to check and if it isn't a matching set of four, that can increase noise too. Ensuring the wheel alignment is correct is important for smooth and quiet running and of course the tyre pressures must also be correct.

Check the door seals and ensure all the doors and tailgate fit properly and are correctly adjusted and ensure all the body plugs are fitted and similarly any grommets and gaiters.

are you talking road noise from the tyres
or exhaust or suspension noises

used tyres matching for noise ...
depends on tread pattern and how much thread
what year they were made

tyre pressure ...sure check all are to spec

pre inspection before any wheel alignment
should will be checking the suspension condition

check mud flaps on wheel arches in case bent and rubbing on tyres

Does the noise only occur at or above a specific speed? I'm hoping for your sake that one of the plastic fender liners or under carriage covers have come loose and are flapping in the wind or against the wheel when you pick up speed? If a tight turn instigates it then my thinking is that the dislodged cover is caught by the wheel at full turn (the clunk) and then flaps and rubs while you drive?

Sounds like a seperation in tire. Inspect outer tread if nothing look for places around tire that not smoothe.

Check your backing plate or called shield on rotor on that side. Possibly, its bent and rubbing the rotor. This could happen during brake work.

Check front tyres & wheel bearings espesially the R/H/F. Feel tyre for high / low spots on inside & jack up & spin wheel if you get noise while jacked up it is not the tyres but wheel bearing or brake pads but more than likly a crook bearing by the sound of it. It needs to be tended to ASAP

What pressures are you running in the tires that are giving you the problem?

low profile tires can cause this droneing noise , or if the wheels have to much air in them , id put 32 in the tyres and try it , if this doesnt work it could be a wheel bearing