Woodworking

The way that occurs to me to do this is to resaw it on a band saw a bit over-thick and then use a wide belt or wide drum sander to take it to final thickness. In moon landing units you want a veneer that's 1/16" thick. I wouldn't trust a thickness planer to that, it would be so flexible it would just pull it up into the cutter head and grind it like a wood chipper. My thickness planer doesn't even go thinner than 1/8" (~3mm).

With hand tools...Maybe make a jig that guides a frame saw - something like a giant miter box, and then I think you'd double-sided tape it to your workbench and go at it with a smoothing plane

Musical instruments have been made far longer than there have been power tools. Methods of cutting veneer with hand saws are documented from hundreds of years. Roubo on Marquetry covers it well enough for your purposes I think. DM me and I'll send you a .pdf.

Where in the world are you located? I have material in yellow cedar- a tone wood very similar to spruce- in pieces 6" wide x thick veneer to 3/8" thick, very tight vertical grain that I can supply you with at an attractive price.

It looks like you're in Europe so shipping may be prohibitive. I'm in the USA.

That is a very thin sheet. You won't get sub-mm precision with hand tools, especially the way you'd have to cut.

You could look for high-end veneer, that could already match your thickness.

1.5 mm with hand tools? The only thing I can think of is a planer to get in the ballpark and then sand. Lots of sanding. Because if you used a tool to do it, it would be a drum sander.

I think you could do this with a pull cut plane after you cut to close to the correct thickness with a hand saw. A pull cut plane will prevent the thin sheet from buckling as you try to thickness it, just have to get clever about anchoring it. You could glue a board to the back and then hook it on a table using that board, plane it until you’re satisfied, then cut the board off.

A good handsaw and years of practice will do it. Cut it to more like 2mm or more and plane it down to your final thickness. But it's not easy! Good luck

Sand paper it down? It'd take ages.

I don't know about hand tools, but a skip planer would work. You could use a well tuned hand plane, but that would require a lot or precision and skill.

My vote is for a precession saw or laser cnc

For very thin veneers, very sharp shaves or block planes with months or years of practice or specialized machinery are probably the way to go to minimize waste.

A table saw with a thin kerf blade is probably the most practical approach, if you don't have months to dedicate to this project. I remember someone doing something similar for a guitar build at a local maker space.