@Dana , two points:
1) Several people have given you reasonable options about how to handle this. You have rejected most of them because they are not what you want to do. But the truth is that you can't enter the Class D without 2-way communication. So you need to avoid it until you do. In this situation, with one radio, I would probably try to contact them on the ground, or on the phone, and just tell them what's up.
I often fly out of HSD, shown below. Many times I want to go east or southeast off runway 18. But because of the Class D, I can't, not immediately anyway. So I make a big right turn, climbing, and usually go around the north side until getting in touch with ATC. It's a few miles out of my way, but hey, too bad, that's the rules.
2) We often assume that whatever we're doing, if it's the least bit unusual, is the first time anybody's ever done something. But that's usually not the case. I'm pretty sure that people departing 1B2 and wanting to head westbound over the shore is a daily, routine occurrence for MVY tower. They probably see it all day long. They probably have their own preferred way to handle it - just call them on the phone and see what they have to say. Heck, there might be something posted on the wall in the FBO at MVY or 1B2.
Two examples I know of in Wichita are at 71K and AAO - 71K being just a couple miles east of ICT in the Class C and AAO where on takeoff to the south you're immediately in two Class D's and practically on final for 3 airports, including an Air Force base. Both these airports have traffic routinely operating out there without a problem, nothing is particularly unique for ATC, and they have procedures posted in the FBO.