Reddit mentions of ProMariner 43012 ProSport 12 12 Amp, 12/24 Volt, 2 Bank Generation 3 Battery Charger

Yes I think getting the batteries to a known state is the best first step. Once they're fully charged by a decent charger you can just let them sit 24hrs and see what the voltage is then. Healthy batteries should settle down to 12.6v after being disconnected from the charger. If they go much lower, or more to the point if they continue dropping below 12.3 while they're disconnected then they're bad and probably can't be recovered.

A trickle charger won't do it though, you need something that has a setting for sla vs flooded batteries and that can drive at least 10A and you need a voltmeter and ammeter, built in is best - some way to continuously monitor the progress. If it's not a quality 3-stage charger then just watch it as the voltage rises to 14.4-14.5 and then watch for the amp draw to fall off. The rule I go by for terminating charging is 5% of the 20 hour amps capacity rating of the batteries. If those are 100AH for 20 hours rated then that's obviously 5A. When the charge current drops to 5A you're done with that one.

Note that most solar charge controllers require that the solar inputs to the controller never be active unless there's a battery connected on the battery side first. I blew up a Chinese controller this way once though not nearly as nice a one as you have there ($12 on Amazon).

Any 10A or bigger charger will work provided it can be set for sla batteries

These following units are cheaper on marine sites. Just google the model number.

A good one:

https://www.amazon.com/ProMariner-43012-ProSport-Generation-Battery/dp/B00F5EBR1C/ref=mp_s_a_1_7

A superb charger for permanent install as generator driven backup to your solar:

https://www.amazon.com/ProMariner-63170-ProNauticP-2420P-Battery/dp/B004S63NIS/ref=mp_s_a_1_4