First, do not overwrite ArcGIS files with ArcView 3.3 files.
I have always installed ArcGIS in a directory named "ArcGIS" rather than ESRI, so I have never run into the problem you describe.
There appear two ways that may deal with your situation ... each is a different way to end up with the situation where ArcGIS 9.x and ArcView 3.x are installed in different directories.
1. Go to the 32 bit computer you are using as the source for the files for ArcView 3.3. Check to see if you installed ArcView 3.3 in a directory named C:\ESRI. If yes, go to step 2. If no, go to step 4.
2. If ArcView 3.3 is in the directory named ESRI, then uninstall ArcView 3.3 on this computer. Create a new Directory named ESRI2 or ArcView3 or IonaraGIS. Then reinstall ArcView 3.3 on the 32 bit computer, but when it asks where do you want the program to be installed, choose the new directory you just created; for example, C:\ESRI2
3. Once ArcView 3.3 is reinstalled in this new directory, follow the instructions in this Thread for copying files to the 64 bit computer but instead of putting the files in C:\ESRI, put them in C:\ESRI2 (if C:\ESRI2 is where you had them on the 32 bit computer). If you did steps 2 and 3, don't do step 4.
4. If ArcView 3.3 is installed in a Directory named something other than C:\ESRI, then try to duplicate this directory name on the 64 bit computer. For instance, if ArcView 3.3 on the 32 bit computer is installed in c:\othername then create a directory on the C drive of the 64 bit computer named "othername" Copy all of the files from the 32 bit computer to this directory as outlined earlier in this thread.
1. Uninstall ArcGIS 9.3 on the 64 bit computer.
2. Reinstall ArcGIS 9.3 on the 64 bit computer, but this time, install it into a directory with a different name than C:\ESRI such as C:\ArcGIS
3. Then copy the files from the 32 bit computer found in C:\ESRI to a Directory on the 64 bit computer titled C:\ESRI.
Following either alternative should result in ArcView 3.3 installed in one directory and ArcGIS 9.3 in another directory and both should work.
Hope this helps.