I would strongly recommend you put your iTunes library on an external drive plugged directly into the computer, not the TC. It is slow and really unsuitable for this use.
Did it work ok before.. like a month or two ago when last firmware update rolled around. 7.6.3 has been pretty much a dog (with fleas).
TC and Mountain Lion have a network bug.. where the OS just forgets where the TC disk is.. there is no real solution..
Need some more info.. if you want to break your neck and try it..
What mode is it running in.. bridge or router or something worse?
Are you connecting by wireless.. or ethernet? If you can and you do use wireless, change over to ethernet.
The problem has really been noticeable in the last few months although it did used to very occasionally happen with my old 1Tb TC, the new one is a 2Tb model.
Except for the slow wake-up I find it's okay for the external iTunes library, I can live with the delays.
Ummm... bridge or router I couldn't be too sure. I'm assuming router as we run the cable modem into it & everything out of it. I set it up but just sort of fumbled my way through.
It's connected by ethernet as are all the remote speakers (AppleTV etc) really only use the wireless for iPad, phones etc.
Your description of the OS forgetting is pretty much how it is. The disc will vanish off the desktop but it will still show in a finder window in the side-bar so it only unmounts from the desktop, not the system.
Setup the TC again..
Make sure the names of everything are set to short, no spaces and pure alphanumeric.
Do not use apple type names with spaces and apostrophe.
Use 7.6.1 firmware.. not the latest which is much worse.
I don't really have a definite solution sorry.. the above could help..
A clock timer that reboots the TC at midnight every night.. might sound ridiculous but unfortunately it is sometimes needed.