I use a conservative duty cycle (15 seconds on, 30 seconds off) which is less likely to cause the 150A circuit breaker in the engine compartment (which works on amp draw over time) to trip.
simple circuit breakers function with a bi metalic strip that bends under heat, and "trips" the breaker. That strip will bend to the trip point at a nominal amount of
amperage. There are time delays from instantaneous to long, but that's only long enough to establish the motor in indeed locked and not just delayed because its starting under load or something. In a properly designed system there is no heat buildup that will trip the breaker outside of its amp parameters that you can avoid by cycling like you do.
SO, if you're running your winch, and its tripping the breaker
1) The winch has issues. It's overheating and causing a spike in amp draw and breaker is protecting it.
2) The wiring is undersized, causing voltage drop, amp increase, heat increase, more amp increase, and so on until the breaker trips.. if youre lucky. That scenario just as often ends in brittle and failing insulation.
3) There's a bad connection causing results like the above.
4) What is likely if there are unexplained nuisance trips unrelated to amp draw, is the breaker being faulty. Replace it. There's a bi metal strip, a spring, a cam, plastic pivot points etc etc etc of some kind in there. Unless they are designed to be switch rated maybe as art of a relay system, they do wear out and they are not precision built instruments. Simply put, you can test 10000 simple breakers and none will trip exactly the same. There will be some far to sensitive, and others dangerously slow.
And another thing, since the tripping is a function of amp draw and not time, constantly stopping and staring the motor is hitting the breaker (and motor) with momentary inrush currents greater than the rating, and is more likely to trip it not less. Stopping and starting a motor is function of keeping it within its duty cycle, and has nothing to do with protecting an over current devise. The same goes for magnetic or electronic breakers which I doubt are in a car.
These issues sound to me like a faulty relay, or one that isn't getting the signal to close, or mistakenly gets a signal to open. Without a wiring diagram who know what spooky way they chose to control the power.