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For economy, I would go with standard batteries and recover them by modern low voltage 3-stage chargers. A new single 24 volt specialty charger would slightly simplify the mod, but two common 12 volt units would actually cost less (about $20 each).
From the sounds of it's needing a specific charge station rest position, part of the problem here is to somehow supply a continuous heavy amperage to the motor which may be traveling in the lifted chair. Running 110vac to a riding power supply (or charger) on a flexible, spring-loaded follow-cable can be a bit dangerous if it gets pinched. Alternately, running full amperage 24vdc on a similar spring-loaded follow-cable will cause a significant voltage drop (therefore current) in spite of huge cables to transfer the increased amperage needed over the entire length. But... if just a 12/12 set of low-volt/low-amp tricklecharge wires must be run over follow-cables, then much smaller 3-wire cables can be used to gently recover onboard batteries.
If the motor is in a stationary drive position near the stairway, then wiring is much safer and simpler. That scene was not the impression I got, however. Still, batteries may be needed for current surge since it is a low volt system.
I would sincerely recommend the simplicity of good chargers instead of a giant power supply. For good battery life, it is imperative that at least 3-stage solid-state charger(s) be used. If the original charger is much more than 10 years old, it is likely a very crude older design. More efficient modern economic examples are small Schumacher designed/built smart chargers sold by Sears or Walmart.
The 3-stage charger units are computerchip controlled to supply a heavier current initially, a lower voltage as the battery reaches full charge and a periodic voltage-monitored on-off charge for storage. They will save your small batteries and make them last a long time. I have some of those for my seldom driven vehicles, bikes, boats and camper. For full size batteries, the $7 Harbor Freight (HF) solid-state battery chargers work ok, but for motorcycle batteries, the HF tend to slowly boil the water off over several months. These HF chargers are psuedo-2-stage in that they limit voltage to 13.8 so charge-current does drop as the batteries reach 100% charge, but the leakage current is enough to still lose water in small batteries. The worst chargers are old transformer driven diode units that overcharge with higher and higher voltage as the load drops off, perhaps like your aging original lift unit.
Modern RV (camper) units usually have built-in 40 amp+ 3-stage 12vdc chargers (converters) and both slide-out rooms and coupler tongue jacks use common 12 v motors that draw heavy current. In spite of the HD power-supply/charger, the motors still need good batteries to function because of heavy peak currents. After all, total parallel winding currents alone determine magnetic strength and therefore torque to prevent stalling.
Good luck.
Wes
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