Upon testing the gap with with this alignment spacer (https://www.thingiverse.com/thing:2267858/), I found I needed to raise the BLTouch by 1 mm. I figured the best way to do that was to take a section of the mount I was already using and create a slice from it in the thickness I needed. If printed in the same filament as the mount, it'd be nearly invisible. Even here, with the mount in PETG and the shim in PLA, you'd have to look closely to tell there are two pieces.
OpenSCAD source provided, so you can edit it to produce shims of different thicknesses as needed. (You could also do the same by scaling the Z-axis in your slicer.)
(Photos are of a 1.8-mm shim made with the goal of zeroing out the probe Z-offset in Marlin. This sometimes kept the probe from triggering before the nozzle crashed into the bed. If I had RTFM, I would've known about the 8.3-mm gap that's supposed to be between the probe body and the bed when the nozzle is on the bed. With the heatbreak screwed in so it's level with the top of the part of the extruder into which it screws, I ended up with the 1-mm shim available here.)