This question comes up again and again.
The Bose Lifestyle systems use 27.145MHz (also used by miniature RC cars) RF (Radio Frequency). I know of no off-the-shelf RF transmitter module for this frequency.
BlueDo sells an IR2RF converter for Bose and they provide a CCF with the Bose codes. It is probably
possible to convert it to an RF2RF converter using either a 418MHz (N. America) or 433.92MHz (Europe) RF receiver module
in place of its IR receiver module.
The RF receiver output is active high while an IR receiver output is usually active low so you may need to invert the signal (or bypass the inverter in the IR2RF converter).