...
I got the display working using panel-mipi-dbi Linux display driver and tried to replicate it in Retropie but it did not work. Is it panel-mipi-dbi Linux display driver available in Retropie 4.8?
RetroPie is not an OS in itself, just a collection of scripts installed on-top of an exisiting OS. So the question is if the panel-mipi-dbi is available in the OS that RetroPie is installed. The RetroPie image (4.8) is based on an older version of RaspiOS, if the display is working on the latest RaspiOS (which is based on Debian 12 bookworm), then you can install manually RetroPie on-top of that RaspiOS version. Support for that OS release is still in beta, but most things should work.
...
What are my alternatives now? I am pretty confused why both displays works with panel-mipi-dbi Linux display driver using almost the same init command sequence (just modify color inversion and orientation) but fbcp-ili9341 driver does not work at all.
I'n not familiar with those drivers and how/why they differ. You may try to ask in the RPI forums, since they probably have more users of the hardware which may offer an insight. As far as RetroPie is concerned - my advice is to start with a working RaspiOS Lite installation and then manually install RetroPie (see here).