I wonder if one could just send them a copy of the UK GAR? I have often done that in the past when pilot and passenger details were requested by some airport.
I haven’t used it myself as at C172 speeds and my 3 hours flight time limit I can hardly reach outside of Schengen from Prague. The AIC link in post #26 sums it up for Czechia. Here’s the current link for the Border Check Request.
Is this Czech GAR still current?