Hmmm, sort of worst case then, but because of this;
I only take the laptop out 3 or 4 times a year.
not so bad.
Basically for the first run you should first set up the laptop so that it uses the same drive designations for the boot/program and music drives (you said you could do this), then do a full installation of MC on the laptop, sync all the necessary files from the desktop to the laptop, then restore a desktop Library backup to the laptop, with settings. I would be doing a manual backup just before doing the sync, and of course MC should be closed on both PCs when you do the sync.
On coming home, you would need to sync all necessary files back to the desktop, then restore a laptop library backup to the desktop.
For future trips, backup MC, sync desktop to laptop, restore the backup, and good to go. You should of course keep both installations of MC on the same version, or close versions.
There are a bunch of file level sync tools available which will only copy changed files, and a few free ones are pretty good I hear. But I haven't used one in ages so I can't recommend one. A quick Google and a bit of reading will find you one. There have been a few mentioned in the MC forums as well.
Now, the important bit; What to sync across? Then answer is mostly in the "Options>File Location" setting area of MC.
The first thing I would do is change the MC backup location so that all backups are saved on the E: drive next to you music folder. I'm assuming you have something like "E:\Music" under which all music files are stored. So create a directory like "E:\JRiver\Library Backups" and change the setting in MC to point there.
The "Cover Art" location should probably be changed to the E:\ drive as well, although if you only have Audio, and store Cover Art either in the audio files, or in the same folder as the files, it shouldn't matter. But it couldn't hurt to move that to something like "E:\JRiver\Cover Art"
But you should all of the settings in "File Location", including the Audio, Images, Video, just in case you have and want to carry some of those files. The conversion cache location wouldn't matter, but could be pointed to the E:\ drive as well.
You should also make sure, if you aren't going to sync some files to the laptop, that you have the
Auto-Import "Fix Broken Links" setting set to "No", otherwise MC will remove files from the library when it can't find them on the laptop.
The last directory that would be handy to sync across is the thumbnails directory, so that you don't have to wait for MC to rebuild all the thumbnails when you switch back and forth between PCs, or have problems with the wrong thumbnail being dispalyed. Thumbnails are located in the directory "C:\Users\[User ID]\AppData\Roaming\J River\Media Center 21\Thumbnails". I've never tried copying that to a different PC, so I don't know if the naming scheme will cause problems. However, you could try syncing that whole directory to the laptop and see if you get all your thumbnails on the laptop immediately. If this doesn't work, just delete everything in the thumbnails directory and MC will rebuild the thumbnails . . . eventually.
So, in summary, sync;
"E:\JRiver\Library Backups"
"E:\JRiver\Cover Art"
"E:\Music"
"E:\Images"
"E:\Video"
"C:\Users\[User ID]\AppData\Roaming\J River\Media Center 21\Thumbnails"
Finally, remember to change the Audio Device on the laptop, and back on the desktop, unless you are taking a DAC with you to use with the laptop. Even then you may need check the setting. Also if you have Media Network turned on for the desktop, turn it off in the laptop. You don't want two instances of MC running Media Network with the same Access Key on your network at the same time, while you are still home.
No guarantees I haven't missed something, but that should give you a good start.