Once the game is installed on the disk, you should be able to play it with any account if you start it using the "old" launcher (located in the Bin folder of the game instalation directory), whitout having to reinstall the game with the ordinary user account.
You can also try to right click the AOGame.exe file (also located in the Bin folder), go to Properties, and under the Compatibilty tab select the "Run this program as an administrator" option.
I don't use an ordinary user account often, but I guess that once the My.Com client was installed with the administrator user account, you can run it with the ordinary user. If so, if you select that option on the AOGame.exe properties, you should be able to Alt-Tab the game even when using the My.Com launcher. But it's just a guess of mine.
Please try this and let me know what you find out.