I had sought advice on this problem some time ago, but was unable to find a solution. It now works, and I hope this may help anybody else experiencing the same problem.
Running on WinXP in a secured network, where the network administrator has removed the everyday users’ Administrator permissions from the computers. Had installed both versions of Access on this computer, and for a while they co-existed peacefully. Then for some unknown reason, they began to conflict such that alternating between versions caused the *other* version to want to re-install itself when it was next run. The workaround was to quickly interrupt the dialog when A2003 tries to re-install itself. Interrupting the installation allowed it to work OK, but corrupted the A97 installation, thus requiring A97 to be installed again. That was a pain because installing A97 required the use of the original installation CDs, it needed to be installed as an administrator of the machine, and the timing was critical — the installation had to be interrupted within about 5 seconds, otherwise the A97 installation would be corrupted.
The long-term solution *appears* to be to remove the MSACCESS.SRG file from the folder where MSACCESS.EXE (A97 version) is located. Actually, that’s not 100% accurate — what I actually did was to preserve the MSACCESS.SRG file in an arbitrary folder, then create an EMPTY copy of MSACCESS.SRG in the executables folder. A97 will not launch if the MSACCESS.SRG file is missing, but a zero-byte version of the file appears to be sufficient to fool it into running.
Now when I run either version of Access, they launch properly without asking to be reinstalled. Hooray!