There are cheap IR receiver/sender modules with UART interface: YS-IRTM.
Basically they are extensions for arduinos etc but since they have a standard uart Rx/Tx you can connect them to a bluetooth-to-uart and talk IR directly from appinventor - without any additional microcontroller in between
first step is to read out original remote commands - here the straight forward way is a BT terminal program - you get 3 HEX Blocks - e.g. 00 FF 56 which is the command of one button.
For sending there is a little trap and i guess many got stuck there - it is not enough to just write the command from the original remote to the serial of the YS-IRTM - every command has to start with A1 F1 - followed by the actual command...
in the end the whole block in appinventor looks like this - just the onboard BT client - no extensions etc: