I was afraid that would happen

As this is for a client and I do not know whether or not it would count as 'sensitive', I am not going to divulge any information. Rest assured that if there was an easier way
of doing this, we would have done it long ago.
@ITman496
Your suggestion sounds good for a one off, but we're talking about hundreds. And they need to be double sided as well. So you'd have to sandwich the spacer between two single side boards, which you'd
have to position very accurately to get proper registration of the top- and bottom layers. Ehm, no, thanks.
@Gyro
Thanks for the welcome. Glad to be on board

Yes, I agree that with so many layers tolerances would mount up. Then again, it also gives you that much opportunity for tweaking. At least, that's my slightly naive take on it, I am hoping someone with more knowledge will chime in and set me right.