A 25 watt light burning for 2 hours = 50 watt-hours...a 20 watt solar panel will be lucky (that's day in..day out..cloudy/clear) to push 10 watts average for 6 hours=60 watt-hours per day. The inverter uses 3.6watts in standby (and switching them off/on a lot is what normally cooks them ...as well as surges) for 24 hours= 86.4 watt-hours. Ignoring the automatic door cuz it's small...the inverter burns far more just idling. Looks like you have about 1/2 or slightly less capability in that 20 watt solar panel....even IF it was on a 35 degree slant to the south to make it work far better...actually just even close to it's rating. Roof top flat would be even better output overall if it had all day sunlight exposure, but the 35 degree tilt to the south if exposed over 3/4 of the day is best.
I would switch to 12volt LED lamp(s)..about 3 LED "watts" would be about right (it's times approx 8 for watts to watts conversion LED to incandescent..times 2 to 3 for LED to florescent). Research (and my #2 pencil/big chief) shows 1-red 1 watt LED lamp (less fighting when cooped up days in a row) and 2-white 1 watt LED lamps (more eggs) with associated 12volt DC to LED "driver"--(1 driver for all lamps...all from
ebay) would do that at only 6 watt-hours usage @ a cost of approx $5 max if you assemble it yourself with parts from China/
Ebay..You could order enough LED's and drivers to make 5 "lamps: for less then $20....maybe another 12vdc timer controller (probably
ebay or your local camper dealers) for automatic operation. But these won't produce heat like an incandescent. Both good and bad?
It's "possible" if using a really lightweight door (not sure now safe from local critters it might be though) to use a 12vdc automotive trunk/door electric lock operator (Caddy trunks have a "kind-of" long throw operator) to operate the door with a simple daylight/dark 12vdc control (there are those about..

) There's probably even 12VDC automatic coop doors out there??
So you could chunk the inverter unless you are just set on using one and have more then enough average day to cloudy day input to run the coop all on 12 volts DC with the existing panel..
Watts is watts... but watt-hours are how ya figure solar power used with time thrown in the equation. It's not enough power being used to try and figure it out in kilo-watt hours like your normal electric meter measures. It would look like .060KW or less.
Steve.