I'm guessing a USB to HDMI adapter uses a driver on the host device to intercept the video signal, convert it to a proprietary format and send it out the USB port where the adapter converts it back to a video signal. In that case, my app would need an SDK from the adapter manufacturer to the convert the data, but even then I don't think iOS apps can send arbitrary data to the USB port.
You could ask Alesis if their dock can play this role of capturing video from the dock connector and routing it to the USB jack. But if they wanted to do that, I think they would have just included a video port.
Bottom line, I would be very surprised if this approach would work.