I've read HD44780 manual, but some moments are not clear, so are there any code ideas, with which I just can simply send required byte to HD44780 compatible screen, in 4 bit mode, but without using LCDOUT statement?
Since LCDOUT works perfectly fine for the HD44780 for which it is meant there's probably never been a need to sort of recreate the same functionallity. Obviously loads and loads of examples for assembly, both PIC and other uC can be found on the internet but you're not interested in that.

What "moments" are not clear specifically? Perhaps we can clear those "moments" up and then you'll be fine.

Are you intending to completely replace LCDOUT or are you trying to find a solution only for the occation(s) when you need to send 254?