Good idea but it is kinda embarrassing when you whip out your phone for them to scan and no matter what they can not get it to read the barcode. Two different apps would not work. I know they could read off the number but thats not really the point of the app.
I don't see how switching apps helps. Either the scanner can scan your screen or it can't. An app won't change your screen's inherent characteristics. Though some barcode scanners seem to work better than others. See if the cashier can try the handheld scanner to see if it helps.
Personally, I just use the option to supply my phone number in cases where I even want to deal with these ridiculous store card programs. Ideally, I just avoid such stores. Can't wait for the HEB in my neighborhood so I can stop bothering with Kroger.
The scanner problem is due to the glass on your device. The laser scanners (Hand scanners) tend to have problems, while the optical scanners (the flat type) can usually read them just fine. At the very least I have my numbers readily available without tons of cards/keychain thingys.
I used Cardstar on my iPhone for over a year and was pretty satisfied. Now I'm using Cardstar on my EVO and in my experience the hand held scanners have always worked better than the flatbed ones. I'd like to hear more opinions, so maybe we can figure out a way to get better results.