I am still thinking fax via computer for the dialing.
clone a harddrive and place in the PC, have your PC make the call, listen to incessant fax beeping for 38 or so seconds and maybe even let the PC redial since the fax didn't go through, swap the harddrive out to the original one. Toss the one used to place the call.
No evidence and your home phone was used.
A couple of people indicated that the CDRs would show that as a fax call.