Yes, it will work fine without the brake circuit.
However, you need to remember that my circuit gets power AFTER the turn signal switch (it is wired to the lights, not the flasher)
If you wire into the flasher, that is BEFORE the switch. I'm not sure that will even work. One side is always hot (orange), the other side (lt Blue) goes to the switch.
If you put the beeper on the switch side and ground to the frame, it could pass enough current to activate the flasher when the switch is not even on.
If you put the beeper in series on the switch side, there won't be enough current to run the lights.
I also read that someone wired the beeper across the flasher, but not on a Savage. That wouldn't work on the Savage. Beeper would get power all the time. It would only shut up if the switch was on and the lights were blinking on. I never tried it any of those ways, so I don't know.
(I stand corrected. Routy did it, and it works great! Don't know where my brain was.. It will beep when the blinker light is off)
If you wire your beeper into the lights, you either need to use the rectifier to isolate the left lights from the right (like I did), or just use two beepers, one for each side.
If any of that is obvious, ignore it.