I do the same thing regularly. I have my master music library on a desktop PC. I also have it on a laptop to use when traveling. There are several ways to do this, some (to me) slow and cumbersome. My method involves setting up MC's paths a certain way, then outside of MC copying files from main to second PC whenever updating is desired.
Here's the high-level of what I do, in case it give you ideas...
1. Use the same drive letter and path on both computers. This allows the MC instance on both computers to find files in the same place. I use a second drive on each computer (internal on the desktop, USB on the laptop), which I set up as M:. I put my music library in M:\DATA\MUSIC. (It is also possible to use different drives but use Windows SUBST command to give a drive plus path an additional drive letter, in addition to the default drive letter.) The goal is to have the library always find the media files (music in my case) at exactly the same location. I find this much better than the methods that involve find+replace to change the path (change C: to D: in your case).
2. Store MC's library files the same place as the media files, rather than where MC normally puts them. My music files are at M:\DATA\MUSIC, in in subfolders created/managed by MC's Rename-Move action (folders named for the Artist but some users do it by Album). The library files (database) are in M:\DATA\MUSIC\LIBRARY. This setup can be created from scratch via MC's Library Manager (create the library then add the media files). With an existing library, copy the entire folder+files structure to the new location then edit the library in Library Manager to point to the new location.
3. One catch is that MC will put some files in its default Windows User location (saved views, for instance), but I've been able to steer all of that to my M:\DATA\MUSIC\LIBRARY location too.
4. Think carefully about where to put cover art / image files, because moving them later is painful. By far the easiest is to store them at exactly the same place (drive+path) on both computers. The find+replace method of editing the image path doesn't work, (no evident reason, but it is disabled for this field). Mine are in M:\DATA\MUSIC\COVERART but there are other ways to store them, such as in Album folders (not suitable for my library). In your case, even if you edited the library to replace C: with D: when you copy library files to your second PC, you won't be able to do this to cover art files. You might have to monkey with SUBST. Or use a technique that stores images in album folders (not applicable to my library so I don't have advice on it).
5. Set up MC on the second computer with the same library name and path. Copy everything to the identical location, made simple by putting library and media files in the same place such as my M:\DATA\MUSIC. When the master PC library gets changes that should be on the second PC, just copy everything over again.
There are small downsides to the one-way copying that don't matter to me. I don't care about Number of Plays and similar. I have to resist any urge to modify the library on the laptop (fix a typo, re-rate a song...) because that must always be done to the library on the Master PC.