Phones have been randomizing and rotating MAC addresses for a long time. With enough antenna arrays you could theoretically track an individual RF source through the store but you wouldn't be able to tie it to a returning customer or identity by itself.
The days of easy external phone tracking are long over.
Your scenario is more than just a software update and dotting some i's. Pulling this off would require a lot of hardware and compute.
The best you could do is force everyone to scan prices through their phone with an app registered to them. You probably won't have many customers left when everyone gets tired of pointing their phone at everything to see their custom price.
But if both me and another person are standing in front of the prosciutto and cured meats fridge, we're seeing the same prices, even if I'm poor and they're rich.
The personalized pricing is usually by having everyone pay through an app. The app knows your buying history and tracks everything you do so they can fine tune their deals for you, surfacing discounts on things that pull you into the store, running e-coupons when it knows you're price conscious, etc. etc.
Both systems are fair on the surface but exploit the asymmetry of billion dollar information systems vs the average consumer. All of these tweaks ensure they get the maximum amount of money that they can out of their customer base which means on average everyone ends up paying more, all while being very hard to point to exactly how you got screwed.
[0] One of the core foundations of neoclassical economics is unbounded rationality which includes the ability to predict the future.
Many places would dream of my "options".
You are making the fundamental mistake of thinking that the current equilibrium of local stores will continue to persist once some of the stores make a deep and fundamental change to their business. That is obviously not the case. It would create a gigantic strategic opportunity for competitors. And competitors really like finding strategic opportunities where they can make a bunch of money now where they couldn't before.
Congratulations!
If you're poor and you're a in store full of millionaires, you'll end up paying millionaire prices, unless it's for an item the millionaires rarely buy.