Chances are you are not going to find an off the shelf solar option with the kind of timer you need for supplemental lighting for winter laying. Most run on light sensors as stated above and come on at dusk and run for how every many hours you have stored energy for...or for the several hours durations you noted. They also may not be bright enough.
You'd need to build your own solar system with adequate collector plate and battery storage sizes to fit your climate. Then timers and light fixtures are available to add on.
Have seen a couple folks on here build these systems over the years.
Supplemental lighting for winter laying needs to be pretty precise and consistent.
Here's a pretty good article on supplemental lighting.
I have one small solar light in my coop however there is no timer on it. Before I retired I went into the coop really early and manually turned on the solar light and a battery operated lantern. I've been looking to find either a solar or battery powered light to turn on early in the morning however, have not been successful. I will be testing some of those battery operated candles that will turn on and off and see if that works. I don't see the need to keep the lights on at dusk. The chickens need the natural progression of the daylight cycle to remind them to get to their roost before the sun goes down.