Cordless Headset - You're not AT the phone when you are trying to barge into line 1, so it's not possible to press the line 1 button.

Does this cordless headset include a dial pad, or is it one of those bluetooth models with a handset lifter?

If you have a dial pad, AND you remove privacy from the voice mail ports (and I guess you already have, since you can barge in from the keyset), dial 6801 when you get the Intercom Dial Tone.