Two copies of Lion installed on one computer, now the App Store won't install apps I have installed Mac OS X an additional HDD.  So my MacBook now has two copies of OS X, and I can select which one to boot when it starts – great.

The only catch is that, if all my hard drives are mounted, the App Store app doesn't work very well on the newer install – it thinks apps are installed when they are not (they are however installed on my old install).  Obviously it is seeing the applications on the second HDD.

I don't want to unmount the second HDD (which fixes this problem, but forces me to stop what I'm working on).  I've tried adding the second HDD's Application directory to my spotlight exclusions to no avail. What other ways can I solve this?
That the App Store application finds the apps installed on your other OS volume is a feature. It may not be exactly what you want in this scenario, but consider the far more common scenario of a user moving some of their apps to an external volume because their system volume is running low on space.

You could file a bug with Apple, but the team responsible is unlikely to be convinced add special support for such an uncommon use case.

I'm afraid you're likely stuck with unmounting the second volume.