First off, if you have 110V AC from any power source, it is faulty and should not be used. The standard is 120V and running appliances or other delicate electronics with too little voltage should be avoided.
Regarding the Thomas Payne recliners, it is surprising that someone with electrical knowledge wasn't consulted on this. If one recliner works fine on a circuit but fails with too, its almost certainly insufficient wire gauge that produces losses (as heat) and results in voltage sag.
The easy solution is to put each recliner on a separate wire run.
The power adaptor outputs 29VDC at 2.0A which is 58W.
Running this at 12V (on a 12.8V nominal battery) would require 58W/12V = 4.8A.
This would be handled with 16AWG wire for the lengths that would be required. This would be a dollar or two worth of wire to make these recliners work without having to have the inverter running.
Its too late for this to be done better at the factory but seems to be a cheap solution to make this better. Running wires on a fully built RV is the tricky part.