Can't determine if since you are using a buck converter to drive the LED, if the 350mA refers to current into the LED itself, or the current the buck converter is drawing from the battery. Either is feasable, but makes a big difference in calculating the power requirements.

One would assume that you are also going to use both batteries in parallel, totaling 12V @ 9AH.

Your best bet for calculating what you need is to convert everything into watts. It makes the math easier.

The very basic raw info you need to know to come up with a workable setup is:

A - How much battery capacity you have, in this case 108WH (12Vx9AH)

B - How much power the LED will use.

Using an average drop of 3.2V for white LED's, 350mA would use 1.12W of power. This is neglecting the efficiency losses of the converter, since in this scenario, that information is missing.

If the 350mA is the power going into the buck converter, you're looking at 4.2 watts, and includes the losses in the converter. Big difference.

C - Worst case scenario of how long you expect to have the light on. Going with the 10 hours you specified, this works out to either 11.2WH (plus losses) or 42WH depending on the above.

For the former scenario, the battery is fairly well matched (and has plenty of reserve) to power the LED on a nightly basis, draining it by just over 10%. The latter, not so much. You'd be around 40% discharge nightly (at a minimum); with not-so-sunny days making the following night pull it well into the 'danger' zone, shortening their lifespan considerably. Don't expect your SLA's to hang around long in this situation.

The PV capacity needed is all about philosophy and how much insolation you get where you live, and different people will say different things. Where I am, an average panel capacity of about 150% of my load covers my use, but I also don't run the loads all night either. On that same note, I usually have a surplus on most days.

If the components are reasonably matched, you probably won't need any kind of special charge controller for this. A 14V zener rated at 1A would provide all the surplus power dumping your system would ever need and then some, and is simple and cost effective.

Hope this helps...

And I will guess the batteries are used, because I hope you did not buy the batterie$ at a mower shop.
All electric start push lawn mower batteries I ever saw were SLAs.
Meaning they were abused from use, and spent long periods sitting under charged or over charging.
Do not expect them to hold 4.5AH.

The 800ma looks OK on paper, but it is close to C/10 of the combined battery rating.
It won't take long before the battery voltage is high, and charge controller will not let that many amps into the battery.  Meaning just because the panel can make 800ma for 4.5 hours does not mean the battery can accept it.  The battery can only supply what it accepted.

I would say you need a larger battery.  Maybe 12AH SLA.  Larger and common enough to be "semi cost effective".

Might find it is about the same cost to get a 85AH deep cycle (that does not claim to double as a boat starting battery) from Wally World.
Leading to you may find a 40W panel is not a lot more cost than a 15W panel.
So a few more bucks gets a lot bigger system.  Run the light, charge cell phones and ipods, maybe a CFL in the living room on summer nights.