Yes, the auto gauge approach will work, and it’s essentially what I use to level my bed after I take my machine apart. The tricky part is that you need to do it with the machine on and at temperature. The heated nozzle being metal will expand and contract slightly with changes in temperature. if you calibrate it cold, chances are it will be too close with the machine at temperature. I find it takes about 5-6 minutes for the nozzle to decide it is done expanding. You do have to contend with small amounts of plastic ooze, which can throw your measurements off. A further complicating factor is the Z height adjust itself can in some circumstances move due to spring preassure or expansion and contraction of the bolt and or the nut. I finally solved that on my machine by buying a cheap small micrometer, cutting the top of the C portion off, and printing a mount for it to my frame. Mine’s a little different than your Taz so it wouldn’t be a direct fit, but definitely something to look at as an upgrade someday. One of these days I’ll get around to publishing one that fits a bearing rod instead of an extrusion.
The thickness you want is about the same thickness as a business card, or one of those little metal combination metric rular / depth gauges.