Need to add blinker module to relay trigger
If the blinker module was intended to be used on a motorcycle, it probably uses the lamp current to heat a bimetalic strip that opens the circuit when hot.  The LEDs won't pass enough current to produce the heat needed to make the blinker operate.

You could add one or two incandescent turn signal lamps to the circuit, on the output side of the blinker module, to give it enough current.