If you simply said "I will betray X store today", I would not know what you meant.
If you said "I usually shop at X but today I will betray X", I might deduce your meaning, but this is not idiomatic English.
I suppose you could write it that way but it wouldn’t sound right. One usually thinks of betray in a more intimate fashion – “He betrayed his country, she betrayed her lover…” To betray usually means to act contrary to a promise or oath that one has taken.
"Betray" is pretty strong. I can imagine "I usually shop at X but today I'm going to be disloyal and shop at Y."
As teksch said, "betray" involves a deep or sworn commitment. Maybe it's a cultural issue, but we don't have such a relationship with our stores in the U.S. Even someone who was a loyal Starbuck's coffee drinker would not consider buying a cup of coffee at another coffee shop a betrayal of Starbuck's, I don't think.
I agree with James, and he makes a good point about disloyalty - in BE we could say I'm going to be disloyal, in that way. It would be a joke, of course, though economists and market analysts talk of brand loyalty; I've never heard anyone speak of betrayal in this sense; to me that suggests more even than reneging on some prior commitment - I associate it with entering into league with the enemy.
For example, I remember using this once in this way, I worked for Tesco (British supermarket) and I was walking with my friends and we walked past another big supermarket (possibly Asda) and I said "Ok, I'm just gonna be a traitor for a second and get something from Asda"
It just implied I had a loyalty for where I worked and going somewhere else was like betraying the company I worked for.
Of course it was a joke, and you needed to know some context, by itself in the way it was originally suggested wouldn't make any sense at all. You'd need to use different language to lower the seriousness of the main meaning of 'betray'.