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It's how I do solid linings. Bend a side width piece instead of 8 individual pieces, then cut them off. I score with a cutting gauge, both sides but my home made scalpel version will do. You don't have to cut all the way through, a fine toothed hand saw will follow the scribed lines.

Yeah that is the one bonus about doing it this way. My binding jobs are pretty s****y to date. I do have a drum sander and I plan to thickness the mahogany purflings slightly proud before installation.

That was my thought with the gramil after it was mentioned. I'll need to make my own if I do it that way. The blade on the LMI Gramil is much too thick and angled to leave a clean cut. The wood is very stiff and brittle (but excellent tap) and I'm a bit worried about running a crack with anything other than a very, very thin kerf blade.

I thickness my bindings in the sander all the time. Thickness is straightforward.

For fine tuning height cut a slot in a carrier board the same width as the binding and slightly less deep than the desired final thickness. Works quite well and the sanded side is pretty true.

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It's not what you don't know that hurts you, it's what you do know that's wrong.

Joey,
The trick with the gramil, or any shop-made tool that is similar, is to go with many light passes. You need to be firm in terms of holding your tool's fence to the edge of your workpiece, but light in your cutting pressure. After several passes, you will have a cut that is more or less self-guiding. Do this: Practice on some thin scraps of whatever gnarly wood you can find in your shop. When you can cut several strips without a wandering blade, then your technique is good. At that point, mount a freshly sharpened blade and take the plunge. Worst case scenario is not all that bad: Worst case is you won't like the strips you cut off. Okay. Go to alternative binding strips on this one and use your grain-matched cutoff scheme on the next one. That outcome would not be a defeat--only a change in plan.
Best of luck to you. And please keep us in the loop. Everyone is rooting for your successful outcome!