How many keys are there and what is the layout?
These days, low-profile keyboards with perfectly flat keys are common. Depending on the layout of your prop, you could perhaps get a stand-alone numeric keyboard or keypad with flat keys and just glue your keycaps onto the flat keys. Maybe you would need to cut squares to put on top to be able to it real keycaps that are hollow inside.
However... key travel is short, standard key spacing is 3/4" (sometimes a little smaller on the vertical, such in case of Apple's keyboards) and keyboards/keypads are often made of entire sheets of rubber/plastic sheet/stamped metal etc. so you would probably need to fit the entire thing into your prop.
BTW. Another hobby of mine is to collect and build computer keyboards. Most of them are of Cherry MX key switches that are now all the rage among PC computer gamers. Cherry also made a keycap-compatible switch module over membrane named
Cherry MY. Those MY modules have some pins and springs that are easy to remove to get flat bases - which means that they could be glued to a flat surface,
and they are just under 3/4" square. I have used those to build a few keyboard prototypes. This could be an alternative ...
if your keycaps have the compatible mount - there are several cruciform mounts that look compatible but are not.
These modules are easiest to get by buying a used Cherry MY / Cherry G81 series keyboard on German eBay. Prices vary a lot but there are a few of them out there for under 10€. (I also have some spares myself I could part with, I guess)
Flat keycaps for this keycap mount (for use as adapters) do exist but they are not that easy to find and they would of course raise the keys.