I'm not an Arduino expert but I noticed there are to things happening when you connect the CBT. First it connects the boot loader (You can see what's happening in the hardware device manager (XP)) which gets its own COM port. After that it load the CBT itself which gets an other COM port.
Because my CBT got stuck in the boot loader it installed the driver for that and afterwards it always got stuck there. Kind of like your describing the connecting and disconnecting. What I did was to uninstall the drivers for the boot loader. Take the CBT out first! Put in the CBT again and wait till the CBT has actually loaded. This can take a few second. Then install the driver again.
This worked for me. YMMV.
Disconnect the battery and measure between the two can lines. Since they're terminated with two 120 resistors parallel, you should measure 60 ohms. Remember to do resistance measurements only with the battery disconnected!
This is a handy page to check some things on a CANbus.