We have a similar job in ABS plastic and made a fixture that clamps the part with compressed air; the unique thing about it is the fixture is 'upside down.
We have two fairly thick aluminum plates bolted together along two opposite edges with a gap of about 1-1/2" between the plates and these bolt down onto the table of the machine.
The ABS workpiece slides in under the top plate and a large diameter but very short air cylinder goes underneath; the air pressure clamps the ABS up against the underside of the top plate.
All the machining is done through holes in the top plate; the holes were machined using the part program with tools slightly larger than used for the part.
Reload time is a matter of a few seconds; turn the air off, slide out the finished part, slide the next piece in and turn the air on.
An open mind is a virtue...so long as all the common sense has not leaked out.