I would try the BGM without a programmed button and just enter #490 01 to turn it on with the phone.

You already verified that the modem works by dialing #19 from a phone so I don't seee why either method would not work. So you created a mailbox of, for example 900, with an extension field of #19. When you call the voicemail and dial 900 (in this example) you should, in a few seconds, hear the modem. I always put an H after the #19 but I would not think it would matter that much.

I know I have used a DID to the internal modem a couple times. I don't see wht that would not work unless the DID is not hitting the phone switch or something like that. You could try to change it to Dialing digits with #19 but not sure if that would work at all.