It's doable, but not pretty.
It involves creating a panel.cfg and setting the size of the registration to zero or nearly zero. Unfortunately, with the last couple of revisions of MSFS, when you have a local panel.cfg, you also have to have a local panel.xml in the same directory (as far as I can tell), or in another mod in the community directory. If you don't, then some instruments may not have their text turn on. It's kind of ok for default aircraft, or where the developer doesn't encrypt their files like the Long Ez and P.149, where you can find the panel.xml and copy it to your livery's panel directory. But you're out of luck for deluxe/premium planes or Carenado aircraft where the panel.xml file is encrypted, unless you can figure out what needs to be in the panel.xml file (basically, what's the name of the instrument and what circuit is it on) which you could get from the systems.cfg file, but that, also, is encrypted.
The other issue then becomes for planes like the C152 that have mods. If your livery loads after the mod, and the panel.xml conflicts with the mod, that causes issues, too. In the example of the C152X or JPLogistics C152 mods, if you have the default panel.xml in your livery, then the radio text is turned off and on with the taxi lights switch (They put the radios on the same circuit that the default plane uses for the taxi lights). In that case, if you delete the panel.xml from the layout.json of your livery, all is happy again because the panel.xml from the mod still works.
Be that as it may, you can check out my livery for C152 N48270 at flightsim.to for an example of how I turn off the registration on the sides and wing of the plane, but still allow ATC to communicate with the registration number.
I have no understanding of why Asobo has made it so hard for people to create liveries.