I tried this, in fact I created some TPU discs and glued them inside the BitJig, but the results were worse, that was my first approach and I switched to the "all-hard-materials" approach
It is JUST like this, the harder the surface, the better results you get
Wood is not bad, I am currently using a 4mm-thick small metal plate that I had. I place the BitJig on top of that, and then I place everything on the floor, or a plastic board (I could use the floor directly, but just to make sure that the nut does not make small marks on the floor). I have also tried without the plate, and the results are quite good as well (in fact that plate is ready to be thrown away, it's just scrap I had, I still use it so I don't mark the plastic boards).
The tip of the engraver/stamp has a “\/” shape (clearly visible in the figure), but due to the hardness of the washer it cannot penetrate deeper into the metal and make the text better readable. Therefore, you can use a file/grinding stone and sharpen the ends of the engravers to make them a little wider (“\_/”), which will allow you to make characters with wider lines.
Nuances:
You will need a 2nd set of stamps and use the 2nd set after applying the 1st set. First, a garver with a thin end - the 1st blow ("\/") is used, and for the second blow it is used with a wider one ("\_/").
I'd say the results are "good enough" with one set of stamps, the words have to be readable, as long as you can read them the do not have to be super deeply engraved (imho). Just like in the video, or maybe a bit more, but as long as they are readable I am fine with it.