I’ve wheeled my bike into and around plenty of shops although I’d rather lock up. Most recently a milk purchase in Sainsburys local, grabbed milk, wheeled to self check out, paid, left.
There was another cyclist in there, astride the top bar, scooting the bike around, wtf. An adult male with a road bike, not a BMX yoof, what a muppet
Also a corner shop to drop off / collect a parcel.
I’ve been refused entry in the past at a Little Waitrose, a Tesco petrol station and got told off at a Shell Little Waitrose.

I’ll only take bike in if for a matter of minutes, not a full shop!! So no need to lock up bike in shop