I have the same trailer as you. 2014 36 CKTS with 4-6v batteries, inverter and fridge. The other day I noticed come corrosion starting on a few battery connections. I removed the battery caps and found the water (acid) level low as well. I will be adding distilled water to the batteries and cleaning & protecting all connections.
Your problem could be as simple as corroded or loose connections. The reading inside might be good but when the battery system is put under a load (such as when the hydraulic motor is run) the voltage cannot travel into the motor sufficiently. When you run the jacks or slides, carefully feel around on the cables. If you find a warn/hot one you likely have found your problem. The hydraulic motor is a 12 volt motor and gets its power from the battery bank. That is why it does not work better when plugged in to shore power. The shore power charges the batteries through your converter but does not deliver power directly to the hydraulic motor.
Since our trailers are the same age and my batteries need some attention it stands to reason that yours might too. Our batteries are too young to need replacing (yet not impossible) I bet it's a poor connection somewhere! Good luck!
Doug & Cindy
2014 36 CKTS: Titan Disc Brakes, EMS, Trav'lr
2016 Ram 3500 CTD 4x4 CCLB DRW: Andersen Alum II, Factory Rear Air Suspension, 3.73