After I broke down the problem, I can try to find coping strategies:
Try to find the product range online.
To counter overstimulation, wear protective items such as earplugs (and perhaps blue light-filtering glasses, which I still want to get), and stim rigorously.
Go at a time where there are less people.
Also plan in time for recovery.
If I can look inside the shop from the outside, discretely observe the place and the people; try to make an "mental map", so that I know where to go.
First time go only quickly, to get a better overview of the place; next time I'll have fewer fear of going to a new place.
Apply hyper-focus and only look for one particular kind of item (e.g. midi-skirts). That means that I also have to make thoughts which kind of style I want to wear.
Remember that I don't need to buy anything!
If I don't have spoons for fitting clothes, buy them and try them out at home. I can later either return them later or donate them.
Avoidance is still a viable strategy. Buying online can be comfortable, since I'm in a familiar environment where I can dim the lights and block ads & animations. I can carefully compare the sizes with my measurements, so that I have to return (and try on) less.