The technical ideas about the microphones sound good. Getting better microphones is always good.
Not sure how you are doing the filming or the editing, but can you do the filming with the regular audio and then go and record the sounds of the working you do want in the background separately? And then add in the sounds into your video using a multitrack audio software like Audacity (free)
https://sourceforge.net/projects/audacity/. Or a top of the line audio editor like Sound Forge.
https://www.magix.com/us/music/sound-forge/. Used along with a good quality video editing software, e.g. Sony Vegas Movie Studio, you can pretty much add in and delete the sounds you need in your video(s).