Frank,

There is at least one logical mistake in
your program. When you send byte via UART
you have to wait time when UART really do
send this one.

When you press one button only it doesn't matter but for several bytes it does!

Please change this fragment of your program
to this one

MOV R0,#RX_BUFFER
MOV DPTR,#UART
LOOP: MOV A,@R0
MOVX @DPTR,A
;
;WAIT HERE TILL UART SEND BYTE
CALL DELAY
;DELAY DEPENDS ON YOUR BAUD RATE
;
INC R0
MOV A,R0
CJNE R0,RX_POSITION,LOOP

Let me know if this helps with your problem.

Be of good cheer,
George