That's not the way my '21 SE winch was delivered from Honda! The winch contactor (reversing relay) was wired directly to the battery. Nothing about the installation was "Key-On" and I consider that to be a good feature allowing use of the winch key-off and engine not running (what if it can't be run for whatever reason? Then key-on for the winch would just wasting power for ECU etc. and possibly screwing things up by powering the ECU with low voltage). This does not preclude the engine from being run while winching, and it fact that's always a good idea where possible so the electrical system can help supply the battery under load.
On my factory installation, Honda got around the issue of accidental winch operation by using a wired remote that you have to plug in to a socket inside the glove box. While this works, IMO it's sub-optimal as the wire is a hassle and is only long enough to get you to the front of the vehicle. To correct this, I installed a wireless receiver and remote - the receiver has a kill switch on the dash so power to the receiver must be turned on before the remote works - again, no key-on power required.
To the OP, if you are going to do the wiring yourself (i.e. not buy a complete harness) what you need is a double pole, double throw, center off, momentary rocker switch (DPDT, MOM-OFF-MOM). This will be wired with the outer 4 poles (two on each end, wired to the two poles on the other end like an "X". Power is applied to the two poles on one end, so depending on which way the switch is activated, power to the center two terminals will be reversed. This way the switch acts both as a momentary on-off and to reverse polarity to the winch contactor depending on which way the switch is moved.