Well hell, I'm board at lunch.
Winter Solstice is Dec 21 this year. At 48.5N there is 8 hrs and 15 min of daylight that day. So 15:45 of darkness. (worst case)
Coop heater: 50W 50W/12VC= 4.16A
There exists
Low Voltage heat tape. Let's assume that you aren't heating the entire pipe, just the part above ground, so like 10 ft? 5W/ft*10ft=50W 50W/12VDC=4.16A
16 ft of LED Strip (waterproof) says it needs a 2A power supply at 12VDC.
Heater and pipe warmer together is 8.32A* 15.75 hours on the solstice=131.04AH
Lights @ 2A for 4hours and 41 min (0400 until 0841 (sunrise on the Solstice)=9.5AH
Total=140ish AH. That's a fair amount of power, actually. A couple of
100AH Sealed AGM Batteries would work. Lithium get's expensive fast.
Now you need a solar panel capable of charging that in ~8ish hours. 140Ah@12VDC is (I think) 1,680 Watt Hours. 1680Wh/8.25hrs of daylight=203 watts (average, for the whole day) KD5NRH probably wants to step in here as there are a ton of variables in panel output, but my understanding is that they are rated in under "Ideal" conditions (that is Noon, at the equator kinda conditions) So you'd need some extra capacity. Maybe 4 of those 100W Harbor Freight panels? Two of
these, maybe? That's getting into $800ish to set up. I did however pick the worst day for solar.