Thanks for your feedback!

I have actually kept digging the datasheets this afternoon and found out that as you say it just needs a LVDS receiver and transmitter! I should have checked your replies before !

In the 20" panel datasheet, they say "Interface chip must be used LVDS, part No. DS90CF383MTD(Transmitter) made by National Semiconductor.
Or used the compatible interface chips(TI:SN75LVDS83)."
It's not exactly the same P/N as you said, but it's probably very similar chips I guess? Though you also advised the SN75LVDS83 too which is said to be compatible in the datasheet.
So if I'm guessing correctly, it just needs to connect as follow :
Host LVDS connector -> receiver -> RGB18 bits -> transmitter -> panel LVDS connector.
It only needs to connect correctly the RGB output from the receiver to the input of transmitter while leaving LSB to GND, right? Following the mappings given in the panels datasheets ? Now the 20" panel datasheet does not give the mapping of the bits inside the channels. But looking at DS90CF383MTD datasheet at figure 2, I think that's the mapping to follow isn't it?
This sounds indeed pretty easy actually. I was kind of afraid that it needs to use a MCU and write firmware.
Now there is still the missing EDID for the 20" panel. As it needs to make a PCB anyway, I can put a EDID on the same board. Though I need to find a reference schematic for EDID.