I think that plans are too far advanced for MIL to easily change and uninvite the other two. And you will be the villain of the peace if you do which would drag on far longer than a day's shopping trip. So you may just have to go ahead with it. MIL shouldn't have invited them as in my experience it will only cause delays in achieving the mission but perhaps they sort of invited themselves?
Your John Lewis idea is excellent and the personal shopper is used to dealing with the shopper's extra people. I once did one in Monsoon which was very good and very affordable.
When my DIL was arriving from abroad and needed emergency shoes. I got a good description of what she was looking for and did a good internet search on what was available in our nearest big shopping centre so we already had a hit list of which shops we needed to get to. We had no time for mail order and had to find something that was in stock and it worked really well ( about an hour online).
You might save some of the time... ( and get your MIL used to the price range in advance so she's clear on what she's probably going to have to pay) By doing some online suggestions..
The MN shopping bit often has great suggestions of online brands that are sometimes in the big department stores ( so you have the best of both worlds, and can do click and collect and try on)
Also gets her used to what's out that.
A separate post about recommendations in the area would be good. If you are in John Lewis, you may as well save time and shoe leather by going to the Place to Eat ( Silliest most literal name for a cafe)
Good luck. It's only one day... and make sure DH has a large G and T and a nice dinner waiting for you when you get home. I would suggest he comes along and shares the pain, but I know in my case that would be extremely counter productive due to the deeply felt sighs and the "buy that one" suggestions that would go on. But he had better be deeply appreciative.