Try saving it somewhere other than the Beat Saber install folder, it has weird quirks when located there (and probably inside Program Files, too) for reasons beyond my comprehension.
I can't test it to see if I can fix it, but I could make some debugging builds to if you're willing to try them out.

I tried it with the .exe on the Desktop and on another drive, but it didn't made a difference. Since the only way to install songs from beatsaver without putting the headset on is to start the game with fpfc and install it in the songbrowser (because the search funktion in BeatDrop bugs :/) I am going to test your debugging builds if you made them. :)

You have to disable and re-enable the OneClick Install options to refresh the location, otherwise it just keeps using the one in the game folder. I'll make a debug build in a bit.

Please try this, and show me what it says.
ModAssistant.zip

Wait, IT WORKS!

When clicking on OneClick Install on beatsaver or modelsaver it tells me that I have to choose an Application to open it with, now "modassistant" appeared in the list and when choosing it, it installs successfully. I don't know what I did that made it appear, but choosing the wrong application was the problem. Because "modassistant" didn't show up, I selected the ModAssistant.exe by myself and that made it not work properly.

Very interesting, I think I have a vague idea of what is going on. Why it happens, I have no idea, but I think I can work around it.

Thanks for your help anyway and good luck with fixing this. :)