some personal experience that doesn't really add up to an answer....
In the mid-1980's I had a flitch of white oak veneer cut to 1/10" thick leaves, intention being to use it for bent-laminated furniture I was developing at the time. 3 decades later the whole flitch is still in the factory crate in an old mill i sold to a brother. sigh.
Observation: at least back then, you could call a veneer mill and have them slice a 1/4 or 1/2 log or larger quantity to whatever thickness 1/10" or thinner desired. The price was fully competitive with the "normal" product, though of course one does not get the same area out of a thickly sliced flitch, as a more thinly sliced log. So while volume cost is similar, cost/sf of face material does increase. I don't think it is directly proportionate. IOW there is less cutting and handling of thicker sheets yielding production efficiencys for the mill, but the drying may be longer or it may be less efficient to modify the cycles for a short run.
Sometime a bit later, (but still long ago
) I had a 1/2 dozen sheets of 3/4" QS steamed beech plywood and one panel on MDF for horizontal surfaces made. The cost was competitive with standard factory made hardwood plywoods of "faintly" exotic character (Like say plain sliced walnut) - about $80 - $90/sheet IIRC. The problem was that was my first experience of the super-thin faces that started to become common. IOW, the material was well made, but it was the thinnest I had experienced 'til then. It had not occurred to me to qualify/quantify that spec. (I faintly hand sanded any scuff marks and promptly shellac'd both sides of all the sheets before progressing with machining & fabrication)
Vague conclusion: You can probably have sheets for a given project laid up to your specs, in quantities as few as 5 sheets, for a "reasonable" premium, including your spec for face veneer thickness. Lost time in the pipeline increasing proportionately. The plywood maker has a lead time, then they have to call the veneer slicer and wait for their timeline, etc.
Almost all millwork panels are custom made, and the standards for those can be high. The formerly common type of installation where you walk into a 5 story atrium and it is paneled everywhere with 4 x 8 (or 4 x 10) sheets of closely figure matched plywood panels with reveals between them.
Besides the wonders of online searching, it could be worth a call to your regular hardwood plywood supplier - they may know someone or occasionally broker a batch.
Here's an example, there's more out there. Absolutely no recommendation, I've never heard of this supplier before Google. But the type of services they offer are typical in millwork.
Veneer Faces - High Quality Veneer Manufacturing in Pompano Beach, FL
smt