Now I changed my strategy to connect a display to the relay CPU...because of the problems switching the E-input using a relay.

Instead of connecting the relay world and the display directly, using an arduino as a communication system between the display and the relay world will be the solution. All control lines, including RS and E, are set by the relay I/O register and read by the arduino. Then the arduino is setting the lines to the display.

That is how the test system looked like:

And ... may be it will work: