I have finally gathered all I needed to test different workarounds, have read about binoviewers on internet, and I have found a solution that suits me.
My first trial was to use different combinations of polarised filters on the bright-side eyepiece and on the barrel of the binoviewer itself. I find it works best when I have a variable polariser filter on the binoviewer barrel, and a linear polariser filter on the bright-side eyepiece barrel, which I can turn to make both sides as bright.
Problem #1: as you can see above, the barrel of the eyepiece is then too long, and I can't lock it in place properly (problem exagerated by the safety undercut barrel, which makes it unsecure and tilted, which in turn screws up the collimation of the bino and renders it unusable)
As a fix, I have dismantled the linear polarised filter, and positioned it in the bright-side eyepiece barrel. I can adjust the variable polariser to give me enough light on the dark-side, and rotate the bright-side eyepiece with the filter to adjust the linear polariser to give me the same brightness as the other side. You can see below the eyepiece with and without the filter in place.
Problem #2: it is great for moon and bright objects, but not too good for fainter objects, where the view is actually better without any filter.
The solution above doesn't allow to take the polarising filter out easily, for faint targets. So I shortenned the eyepiece barel to make room for the screw-in filter. It takes some time, but with good quality sand paper (the type that can be used wet), and an hour of grinding, the eyepiece is now 5mm shorter, the thread is still working very smoothly (even better than before), and I can easily screw in the polarising filter as I wish, and remove it when I want.
Here is a picture of both eyepieces after grinding 5mm off one of them, and the resulting fit:
DISCLAIMER: I have properly and carefully dismantled the barrel, the optics inside it, made sure to grind on a flat surface. I wouldn,t recommend you undertake this modification if you are unsure about what you are doing, as you could easily ruin the eyepiece.
Hopefully this can give ideas to someone who is bothered by the difference of brightness between the 2 eyes: some people easily adapt to it and don't seem to be bothered as much as me, but I really couldn't enjoy the moon views.
P.S: I also tried to find the right ND filter that blocks just the right amount of light (spoiler it's impossible), put it on the bright-side eyepiece once and for all as above, and then my binoviewer will be perfetly balanced and I will be a happy man. Unfortunately, after trying ND 0.3, 0.6, 0.9, and realising that they all have good results for some targets and less good on other targets, I questioned the fact that the binoviewer is always unbalanced. And indeed it is not.
I set up a camera to measure the brightness of each side on different targets. I discovered that the angle of the binoviewer (more or less closed) and the source itself makes it either balanced or massively unbalanced. This is due to polarisation of the incident light, and polarisation happening inside the binoviewer itself, due to reflecting prisms.