Open your wallet: you need to upgrade. The alternative: run AE 7 on the old machine.
I also recommend making the purchase of the latest version of AE -- CS6, aka Version 12 -- very soon if you want it. Adobe is completely revising the way people procure software, and I can almost guarantee that the occasional or rare user as you appear to be will 1) gape in disbelief, 2) scratch the head in confusion and then 3) get EXTREMELY angry.
Is there a way I can get the install to work,
No way. You can't fix or circumvent what Apple doesn't provide the technical basis for.
As the others have said, there is no way to use After Effects 7.0 on Mac OSX v10.7.
After Effects 7.0 was written for the PowerPC Macintosh, before Apple even had Intel processors. After Apple switched to Intel, they did include an emulation layer (Rosetta) to allow PowerPC applications to run, but Apple stopped including that in the most recent operating systems.