Sounds like an ideal situation for a solar panel. Most solar panels will charge 24 volts.
One reason for not cranking it unless the batteries are charged is that the amperage the starter draws seems to be very high in that situation. This can resulit in the starter solenoid contacts welding together (starter will be "stuck" and you'll have to hit it to jar them loose or disconnect the power from the batteries) Also you almost always have to crank for longer than normal, so more heat builds up in starter windings/armature because of that as well.
But the question of how charged the battery is isn't easy to answer. So, you usually try it and if it doesn't turn over well, stop and use jumper cables or charge the batteries before you do some damage.
I think you will be OK with 2 twelve volt chargers, one on each battery as long as the charger case is not grounded to either output lead. This is because on a series connected setup like your tractor, one lead from both chargers is connected to the same point (the jumper cable connecting the positive of one battery to the negative of the other one). If the cases of the two chargers are grounded to the negative cable, that would create a short circuit if both cases were touching metal or each other. I haven't seen a plug-in trickle charger that was grounded that way and all of the ones I have seen are in a plastic case, so no problems. The solar charger might be able to keep up 2 batteries at 24 volts, but I am betting that 2 separate chargers would be much more satisfactory in the long run just because of the increased capacity. The main thing would be to spread them out so they get good sunlight. With long enough leads, you shouild be able to angle them to catch the maximum amount of sunlight for the time of year.
Batteries differ in their self discharge rate. Good, new batteries will hold a charge a long time, older ones may be dead the next day. It all depends. You will have to try it and see.
It is very likely the battery that was not being charged has become sulfated and will no longer accept as much of a charge as the other one. You may be ahead to purchase a set of new batteries before you setup your trickle chargers. This is the best way to have the batteries charge evenly and discharge evenly resulting in maximum life. It is usually false economy to replace one battery in a series string because of the internal differences in charging rate, capacity etc. Have an uncle that always tries this on our tractors. As a result, we are ALWAYS buying another battery to put in something. I will put a used battery in series with another used battery to get some more use out of them, but only ever put new and used together in an emergency and then only on a temporary basis.
Make sure the tops of the batteries are clean-if they have a film of acid on top, it will conduct current between the posts & the battery will discharge just sitting there...when it's charging hard, like being jumped, acid vapor will come out the vent holes & get on the top...may be helpful to disconnect the batteries if the machine is sitting most of the time, to eliminate any draw from the wiring, etc...Like was said above, a solar trickle charger sounds like the ideal thing...
Almost all CAT's have a "Battery Disconnect Switch", when not in use, you should be turning this off to prevent un-authorized use.
As far as charging series connected batteries individually I have had no problem keeping a matched pair alive. I've been running a pair of Interstate batteries for twelve years and just now facing replacement.
By definition a "trickle charger" is something you would leave plugged in continuously or at least days or weeks at a time to maintain you batteries fully charged. Even a 10 amp charger would take many hours to fully charge a battery. I usually figure 12 to 24 hours because they taper down as the battery charges up. I doubt you are running your generator long enough.
You could take your batteries out and take them to where you have power and fully charge them and go from there.
I do not have any direct start Cats but I would think if your batteries are good and fully charged you should be able to start the Cat several (many?) times over a weekend.
When I say solar panel I do not mean the little ones they sell as battery maintainers. I have a 15 watt panel I use to keep a battery up in an old truck. I also use it to charge the battery for an electric fence sometimes. I know it is small enough that it does not need a regulator. I would look for a bigger one, you may have to do some research to find out how big you can go without a charge controller, or maybe a charge controller is the way to go. A bigger panel would not have to be in constant direct sunlight to put out enough juice.
I know if I had a direct start Cat and the charging system failed or did not have one I would just go ahead and charge the batteries. I think even OM has hinted that he is doing this on at least one Cat.