"If it's the case that the panel has to bring the whole system up to 13.3v to charge, then how exactly do small trickle chargers ever work on larger battery banks?"
At a certain point, they don't. A trickle charger needs to exceed the voltage of the battery, and to supply sufficient amperage. For a Group27 battery, 30-50 milliamps may be enough. For a pair of T-105s, that will not be enough and you may need a 250mA charger to trickle charge them. (I'm not looking up the exact figures.)
The charger--any charger for any system--must be matched to the system requirements, or it will either undercharge or overcharge or waste a lot of money.
"When you are talking about charging voltages, are those with the panels connected, or unconnected? " The only numbers that matter are "connected" with the device in circuit and under load. That applies to solar panels and chargers of all types, alike.
It is also possible that your trcikle charger would be fine--but your batteries are old and sulphated, and they will no longer accept the charge the way they should. In that case, if you test the batteries without a load, you will only read the "surface charge" on the cells, and that drops dratically as soon as you put them under load. Or, if they are allowed to sit for 24 hours without any charging, and distribute the charge internally. (Yes, a charge may have a physical "location" in a wet cell, as the electrolyte and ions slosh around and migrate through the plates, or not.)