It stemmed from your statement that air has no weight. As to whether it's relevant? Hmmm...
Remember one thing. Weight of anything is only relevant to what we read on a scale or what we feel as a force if we pick up any dense object. If you hold your hand out, you never say that you have any weight in it and for that reason air to you weighs nothing until you actually see the effects of that nothing. Basically it is NOT relevant to this experiment, at all.
You would have to allow for it in your experiment, as it could skew results slightly.
It won't skew any results as at all. The scales are inside the chamber and the air is pushed out of the chamber so it totally negates it. It's the scales and object in a partially evacuated chamber, meaning a small insignificant amount of pressure.
The scales before you do the evacuation are calibrated to the additional weight of the air inside the chamber (though it would be small). Obviously you cannot, nor should you recalibrate scales after you have evacuated the chamber. For this reason, you must use scales that can read a negative weight, to cover all eventualities.
Once you put the scales inside the chamber, you will note 1kg (for instance) plus the weight of your object upon that scale plate. you can see that through the chamber before you evacuate pressure, so you know what it reads.
You would first need to do the experiment without your block of wood. You would need to put the scales in the chamber, calibrate to zero and then evacuate the chamber to see what impact is has upon the scales. A control as it were. Then repeat with your block of wood - calibrate to zero in normal environment, put block on scales and then evacuate. You can then account for any differences from your control experiment.
You can do various experiments, as you say. You can put the scales inside the chamber set at say, 1 kg, then see what happens to the scale on its own when you evacuate the chamber. Whatever the reading says, you then know what to add to the scales in calibration to bring it back to 1 kg under evacuation once again. From that point you can then put your object inside and evacuate and read the difference.
The other way is to simply put the 1kg added scale and objects weight into the chamber and evacuate.
Just vary it all with different objects of different densities and log it all down, then let's see your results. I'm not after arguing with you, I'm only after gaining honest results to tests. I think we all deserve real answers.
Personally I would expect different results than you would, from our different beliefs. I would expect (backed up with anecdotal evidence) that actually the scales would INCREASE slightly in the control experiment and the block of wood would also weigh ever so slightly more, as air creates a slight buoyancy - lack of it would make things record a higher weight, albeit a very small increase.
Then this should be an interesting test for you. Good luck.
Conversely, you would expect your results to show a decrease in weight as you believe atmospheric pressure pushes things downward, creating "weight". It would be a interesting one to carry out - I even found myself asking about getting access to some equipment today, but I think it's unlikely.
Try your best to gain access if you can. Equally you can try an alternative method if you live near a sky scraper that you can access the roof from and take a set of scales and a few objects up there. It will be minor differences but a small set of delicate, digital scales would aid you.
Lastly, we may differ in what we think - but I don't get narky with you or anyone else, I maintain civility. Do you think you could now please stop with the veiled insults you have thrown my way? To mind so far, weak, tedious, arrogant, game playing, "sad as hell", and having attitude. I am not interested in trading insults. Thanks 
I'm not getting narky with you. I'm simply trying to keep you thinking for yourself and hoping you don't get swayed by the others. I think you have the potential to be a good flat Earth theorist, as long as you stay strong and use your own thoughts. All you need is to open your mind a little bit and you will see how mainstream science has moved to manipulate your thoughts. We've all had it done to us, it just takes a strong mind to look past it and see a different take on how we are told things are.