Modify generic chinese LED driver circuit
Your S1 is almost certainly not an closing switch, 







simulate this circuit &ndash; Schematic created using CircuitLab

but a momentary pushbutton,







simulate this circuit

considering it's used to cycle through modes.

This sounds very much like 


  Throw away existing circuit, replace S1 by toggling button, be done.


Alternatively, you could also 


  Throw away existing circuit, replace with circuit that toggles output based on push button.


But that'll require a couple more components (two transistors for toggling, plus a bit of capacitance for  de-ringing  the pushbutton) or a small microcontroller (e.g. an Attiny8).

Modification of the existing circuit is not really an option – it's almost certain that the IC is single-purpose; I can't read the datasheet in Chinese, but seeing that it's only 2 pages in length, it's certainly not a programmable controller.