My understanding is that even when driving, the van alternator puts out less charge than the 2k inverter consumes so if your battery is dead(or moribund) the alternator alone will not be able to charge the inverter.
While I would guess that your battery is dead, I would try to "recondition" it before throwing it away. On my Tripplite inverter, there is a 3 position switch on the back; On (regular), Off, Direct pass through to the battery.
Try leaving it in Direct pass through for a couple days and see if it holds a charge any better. If not, it's toast.
My 2008 installed battery from SMB West is
from East Penn / Dekka Manufacturing Co., Inc.
The part required is a Dekka 8A4D-T975 because it has slightly different terminals than a normal battery. It was a special order for the Regional East Penn depot and mine (in Washington) came out of Arizona.
I believe the retail price was $330 w/o install.
My lesson has been to use the inverter sparingly because it seems to eat the battery even without any devices being attached/used. The 12V is much more energy efficient as are smaller inverters for items like charging cell phones, computers, etc.