Its not leaking from the black fitting when running to waste as you have less pressure running through the membrane.
Once you’re turning the tap to start producing, you’re ramping up the pressure which looks like it’s causing the leaks.
Personally I would say it’s not seated correctly, although I’m not sure why it would work the first time and not the rest.
Either that or something is now restricting the pure causing back pressure.
If it were me (and apologies if you’ve already done it, I can’t be bothered going back over all the posts), I would take the membrane out, rest set it and the reset the end cap correctly with the rubber “o ring” it should have.
Maybe give the brass fitting a quarter turn as well to tighten but be careful not to over tighten, especially if the thread it’s screwing into it plastic.
Those valves are crap btw, be worth changing it to a gate valve if you ever get the chance.