but might 36v from a pair of panels damage the actuator circuitry?
So here's the deal. Lead-acid batteries look electrically like a voltage source/sink with a small series resistance, with the voltage level a function of state of charge. 2V/cell (there are 6 cells in series in a 12V battery) is nominal, and if I remember right, their open circuit voltage is something like 1.9V empty, 2.1V full. That covers 90% of their behavior.
Considering that, the "1W@18V" spec of the solar panel isn't going to be able to "win" against the battery, and the solar panel's voltage will be pulled down to battery voltage, delivering probably 0.055A (=1W/18V) at whatever the battery voltage is.
When a battery gets completely full, however, its series resistance goes up dramatically, and the voltage goes up, until there's enough voltage to start electrolysis of the fluid and you get H2 and O2 generation at the terminals and loss of the electrolyte. A lead-acid battery, depending on the type + manufacturer, has a certain recombination rate of H2 + O2 => electrolyte that it can handle; if you electrolyze at a higher current than that, it leads to permanent electrolyte loss (+hence capacity loss)
So there is a safe current that can be delivered to a lead-acid battery continuously, where its own self discharge due to electrolysis balances the charging current. It depends on the manufacture + construction. I wouldn't feel worried about a C/10 or C/20 rate of charge (where C = the current needed to discharge a battery in 1 hour). Garage door batteries are probably > 1Ah capacity so you should be safe with 55mA charging current.
HOWEVER -- I would probably put a (zener diode and resistor in series) in parallel with each battery, the zener diode being about 14V and resistor being maybe 10 ohms or so, so that it keeps the battery terminals from getting charged too far.
Also: if you can, wire each solar panel to each battery (and keep the diodes), rather than the pair of panels in series wired to the batteries in series -- i.e. try to connect the center taps. By doing so, you'll charge each battery independently. Otherwise, what can ruin battery life is if the battery voltages diverge -- the one with the higher voltage will tend to get overcharged, while the other one will tend to get overdischarged and not completely charged.
Your approach is basically correct, but I question some of your numbers:
How does "1 day of reserve" only require 30% additional capacity, rather than 100%?
You've neglected to take into account the storage efficiency of the battery itself (Wh in does not equal Wh out), which is only about 70% for lead-acid.
For your 275 mA × 9 h = 2.5 A-h daily load, you need a 10 A-h battery, which accounts for your 100% functional reserve, plus 100% "battery reserve".
Your charger will need to supply 7 A-h per day, which is the functional capacity (including reserve) divided by the storage efficiency of the battery.
If you have a very good charge controller, it might achieve 90% power efficiency (but possibly much less). 7 A-h × 12 V = 84 W-h nominal.
84 W-h / 0.90 efficiency / 4 h (equivalent insolation) = 24 W panel.
Best Answer
With a conventional solar setup, solar panels charge deep-cycle batteries. A charge controller is used to prevent overcharging and correct voltage and current. Car batteries are for a surge of current to start the vehicle. I'm not sure if a conventional charge controller could charge your battery correctly because deep cycle and car batteries have different chemistry. But from what I found Here, you should be able to.
I would just try it with a charge controller, or try to make one. Just experiment with some different setups and find something that works for you. Also, I don't know if 5w could power your battery because it depends on the capacity and the current draw from the Arduino.