Yes, this part terrified me also! And was all new to me.
I also did not like the Stewmac bridge suggestion, or the idea that the strings should be just under the coverplate. Those ideas did not work with my build. Definitely, the string were not touching (or near) the underside of the hand rest.
I'm not sure how much the cone/spider/bridge assembly compressed with strings! When pressing down on the cone I thought it moved only a few mm.
I feel that you may want to shim the lower end of neck socket a touch to increase the neck angle compared to the body. Maybe try a 1mm thick shim (or some plastic to test the idea). This should get the straight edge (resting on the fretboard) to be a bit above the bridge support, and lower the action when the strings are on. But check with the strings on again! Lots of trial and testing needed! Maybe you should order a couple of wooden bridges.
Looking at my wooden bridge now, see link, it is sitting quite low in the support - and my strings are set very high.







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