I think that’s a cool idea. I don’t see the problem with letting them know about it. It is none of their business what you do professionally, but since you are considering publishing this in your portfolio it would be wise to have their blessing. And I would phrasing as just that. Maybe write them a nice letter telling them why you chose to sample their business for this pseudoproject, and that you’d like their blessing. Who knows, if they’re cool about it they might actually employ you to implement your proposed redesign
In any case, it would be great if you could follow up with us and let us know how that goes. I’d be interested to know how this works out for you.
All the best, @narekk