Will there be much of a benefit from getting the Bushnell 4-16 x 50 over the 4-16 x 40?
I'm looking at either of these units or the Black Diamong 4-16 x 50.

It is for a 243 Weatherby that I use for longish range fox work either lamping or daylight. The intention is also to use for Dawn/Dusk deer stalking

All other things being equal, a larger objective lens will theoretically produce a slightly brighter image. Your own eyesight can play a major role in exactly how much brighter the image may appear to you.

A scope with a smaller objective, but better resolution, can oftentimes produce an image that appears brighter than one with a larger objective but lower resolving power.

At 16x you are talking about an exit pupil size difference of about .5mm between a 50mm objective and a 40mm objective.