I have a similar problem with one of my laptops.
The Modem is not always recognized, right away.
This is my work-around for that.
If the Modem shows up in Network-Manager great, other wise . . .
I have to log out and boot into a LiveCD of any Mint/Debian OS-distro and 'set-up' a new connection.
Then, I can log out of the LiveCD and Log-in to the installed LMDE.
Usually, works fine after this ... and it does not drop the connection.
The Modem is then, usually 'seen' by the Network-Manager.
Don't ask my why this works, but it does.