This is probably a silly question, but in the pictures of a Hyperdrive setup I can find there's wide ribbon cable connecting the controller board to the clip that goes on top of the CPU. It looks like in some of those pictures that you should be able to unplug that cable from the board. If that's indeed the case then I'd probably suggest the following approach:
1: Get the Brainstorm working in a 512k. Physically that's going to involve soldering the machine pin socket to the original CPU, as in the Plus installation I saw and, as noted, figuring out the appropriate place to anchor that flying resistor. (Anyone have the installation manual for a Brainstorm sitting around somewhere?) An interesting question is of course whether the control panel for it would work in an OS version supported by a non-e 512k, or if you'll need a 512ke.
2: Once you know the Brainstorm works you'll proceed to design a replacement for the CPU clip that consists of a PCB board that has through-holes for a 64 pin socket and routes all the pins to a header you can plug your ribbon cable into. Leave enough clearance for the Brainstorm between the socket and the cable header. Order it from your favorite PCB supplier.
3: When you get it, solder ribbon cable header pins and a *wire wrap* DIP socket, IE, the one with the really long legs, like this:
to the board. Plug the Brainstorm into the socket, plug the whole sandwich into the machine-pin socket you soldered to the original CPU, re-attach the Brainstorm's flying resistor, and cable up the Hyperdrive. And that's how you do a turkey.