Mowers have tiny little alternators.
Some have a 'large' magneto, some use the starter motor.  Probably other types too.
No mower expert here.  Even if the neighbors think I am.
Couple local guy's have about 12~16 HP lawn tractors.
The charging systems do not keep up. They use a battery charger between uses.
I would be kind of surprised if they make 2A more than the motor needs to run.

I highly recommend having a back-up generator from the start to save your RE battery from being deeply discharged. Its my understanding that if you are now connected to the grid, you can use that to top off your RE battery pack when there is no wind/sun.

If you have no grid, a back-up generator is a vital piece of equipment (IMHO).

That being said...first I would get a large chest style freezer (opens like a laptop computer), and attach additional insulation. Put a couple of jugs of water in there, and during an outage the block ice will extend its usefuleness.

Bar-be-que some freezer meat the first day of an outage to make room in the freezer for some of the more sensitive or expensive items from the fridge that are slowly warming. You can even go "old-school" and put one of your ice blocks on the top shelf of the fridge to help it last longer.

Concerning the lawn mowing tractor; lawn mowers are thrown away on a regular basis. They typically have a clogged air filter which leads to a rich carbon-fouled plug. This results in hard starting and difficult pushing from the dull blade thats never been sharpened. After 3 years or so, no one pays $50 to have someone fix it when they can get a new one for $150, so, fixable mules can be found in the trash for free.

There's a dozen examples on the web where someone has taken a free lawnmower, fixed it them selves, and attached a pulley set that spins an alternator to provide 12 volts.

Your riding mower has a bigger engine than most push mowers. It would not be difficult to temporarily remove the blade and attach a pulley set (in an emergency) to run a fairly large generator head.

The famous Amatek (and other Permanent-Magnet-Motors / PMM's) are fun wind projects, but are not huge power producers. Perhaps a similar PMM/3-phase servo motor would be very useful for this purpose.

If you think this would work for your situation, of course you would need to fabricate mounting brackets with a control panel and then test the output, so the set-up would be standing by when such an emergency hits.

Here is their page with brackets for those small lawnmower engines.
http://theepicenter.com/tow02077.html

You could either buy one of their brackets, or look at it and make
one yourself.

If I get the gumption to hook it up, I will put a 50/50 Ammeter inline and see just how much the mower shoots into my batteries (a pair of 6v 200AH Sam's Club golf cart batts).

Then I'll do the same with the car.  

But my original post was questioning if a lawn tractor was a more EFFICIENT generator.   e.g. does it burn less gas sitting there in idle?  I imagine I was right about that because nobody corrected that part.

i doubt that engine will burn a half gallon per hour charging maybe 30-40amps
but do get a 100amp or larger unit if possible,, 30-40amps continuous on a large alternator will run forever without problems. and if you need more amps just throttle up the motor and crank up the reostat which is very acceptable for quick charging if you keep an eye on the batteries and don't overheat or overcharge them, which probably is not likely in an emergency situation and the batteries are pretty much drained anyway.

i don't think there is a car on the planet what will idle while producing charging current and use less fuel.

also i you botch the connections when it is dark and rainy or your tired you end up toasting the add on alternator on the tractor, something you can replace easily ourself, toast your car alternator or do other damage and you have an expensive repair and your on foot maybe,,, or maybe even a tow bill.

my vote is an add on to the garden tractor :)