Your processor upgrade is plug-compatible with the stock processor. The G4 turned your Beige G3 into a faster G3. The beige G3 uses the Toolbox ROM, aka Old World ROM. It was not updated
per se. A patch was added to the System folder to enable some of the features on the G4. The 768 MB RAM restriction was not a limit imposed by the G3. It will not be lifted by the G4.
BTW, the maximum OS supported by the beige G3 is MacOS X 10.2.8. If you want to install a more recent version of MacOS X, then you will need
XPostFacto.