Indeed they did supply the 350 EW with a sidecar. As well as for the 2-3/4hp CW that preceded it, and the A31 model that followed. But if you have experienced the massive reserves of power of the solo 350EW, then you will know that the acceleration as an outfit will be underwhelming.
Well heed the factory caution to fit a smaller sprocket to the gearbox. Unfortunately the rear sprocket on the 350EW (if it retains its original wheel) is soldered to the brake drum which is part of the hub. So changes in overall gearing are restricted to the transmission sprocket.
If you go this route, I can get you some information for the bracket that clips to the rear chain stays (item B in the plan view.) I could get you something on the later A31 chassis, which probably changed very little from the 350EW from a few years prior. I also have a 1930 big twin chassis, which likely shares many similar features though it be heavier; it certainly is of a similar design. See picture
here.
In turn, if you come across a hub and spindle, I could well use a sketch of it to replicate one for my sidecar.
-Doug