I started looking in to this and then saw a not so well advertised unit on Ebay and won it for just under £200 quid, it's one of the older blue cylindrical types with some blades although mostly big flat ones.
But I will share what I found and my ideas because who knows how long it will last and it's all good fun anyway.
What's inside a Biax:Biax 7EL assembly/repair:For some pictures of the assembly.
And here is the patent for the variable stroke drive:
https://patents.google.com/patent/US2940324A/enIt's a relatively simple mechanism, the adjustment changes the angle of bearing with respect to the axis and a connecting rod with bearings couples this to the output which is constrained suitably with a linear bearing or way.
Fundamentally you need a motor, gearbox and this mechanism.
Inspired by this angle drill which is clearly an add on:
Makita angle drillI suggest a geared drill with metal body, remove the chuck and clamp scraping doodad where the handle normally goes. Something like:
Makita rotary drillIt has a max rpm of 1200 which is lower than a biax but OK I think and the weight leaves some overhead for the mechanism, ergonomics might not be ideal but not sure. I imagine a DIYer could live with it or get the saw out.
Cheers,
Graham