I'm not sure how much extra the charge would be because roaming probably will be involved, but it has to be cheaper than making a ship to shore phone. I'd contact your cell phone provider and see if they have an agreement with DCL.
I would use the online form from the member website the day that you board and I would explain that you are on the Dream so you may be a day early with the request. It takes MS a day or so to process the request so the timing might be OK. I would check your email from the ship or log on to your account to verify that the reservation was booked.
I was on the Disney Magic when they started taking reservations for BLT before it opened. I used my cell phone from the ship. Hope that you don't have to wait on hold too long because the charge is something like $3 per minute.
For High demand options, having someone added as an associate and calling on your behalf is the best option by far. Otherwise send an email after MN on the day the window opens and hope for the best.