I faced similar problem before (long time ago), and in that case I turned out using HTTP binding, and thus composing the message myself to be able to do all the quirks to the message. It is not a very elegant approach, but doable. I had to say that when I did it it was not with SwithYard, but with other product, but it should work the same way.
The way to go should be to create a custom MessageComposer, ContextMapper that extended the Http based ones, and that your binding would be Http. Probably that could make it work.
I would recommend you to have a look at:
This is where all the magic happens, so look on how to get the magic from the SOAP one into the Http one, in a Custom Message Composer and Context Mapper.
It would be great, if you make it, if you could share some details back to the community.