What made me think this location is "private" was not the word private, but the fact that the mount point is in a container and there in an invisible folder (name prefixed with a period). It _could_ be the intend of the CloudMounter software to hide this from Spotlight. The reason for this _could_ be that the files are not actually available, but only downloaded on demand.
Then again, I really don't know how CloudMounter works. You should ask its developer about Spotlight indexing.
You can check if Spotlight sees this as a volume. If so you can try enabling indexing using mdutil and mdimport commands in Terminal.app
1. Open /Applications/Utilities/Terminal.app
2. Paste in the following line and press return:
mdutil -s -a
3. The command will answer with a list of volumes and whether indexing is enabled. For example:
4. If you see you cloud mounter volume there, copy its path and use it in the command below to enable indexing. Copy the command. Correct the path. Press return:
mdutil -i on "/Users/FritzB/Library/Containers/com.eltima.cloudmounter.mas/Data/.CMVolumes/Private"
5. Check again using "mdutil -s -a". It may not be possible to turn on indexing.
6. Wait a couple hours
7. If indexing is on, and search is still not working, you can try to force indexing to start using the following command: