First, I need to admit that I'm really short on info on the seat! That leaves me very much doing some guessing and you may find it totally wrong---just the best idea I have until somebody might come along with better experience on it?
What I would expect, but total guess, is that there would be a 12VDC motor which runs forward or back when the polarity of the power changes.
If you are able to get the switch out to look at the rear, it may be easy to spot if I am near right!
Since it works to go forward but not back, I would guess the power is getting to the switch and from the switch to the seat fine but not when the switch is pressed to reverse the power. Most of the wiring and motor are good, just not when it should reverse.
Whether you can get the switch out and see the back side may make a big difference in whether this is something to try or not!
Back pain from driving or back pain laying in the floor??
What I think is likely is there is a set of six contacts/lugs on the back of the switch and inside it. Rather than the motor wiring failing, I would guess it more likely the switch contacts are not all good.
This is what I might think you would find looking at the back of the switch, based on guessing!
If you've been a model railroader, this will sound familiar as a track reversing switch!!
When pushed to recline, the center contacts with positive and negative battery on what I've labeled 2 and 5 are making connection to the motor and it turns one way to recline. Pushed to the right as drawn here.
But a DC motor of this type runs in reverse when we change the positive and negative feed to the motor.
To go the other direction, you need the positive and negative to go to the motor but reversed! I'm guessing that contact is not getting made for both wires, so if you take a small wire or anything metal to connect the two center lugs to first one set of outer lugs and that moves it forward, try connecting the center to the other outer lugs to see if that won't make the seat move the other way?
Warning! This WILL blow a fuse or breaker if you touch the wrong lugs together!
So maybe a safer way is better-if the connections are right?
If they are push on connections, it is much safer to find the two wires that go to the motor, pull them off and reverse just those two wires. That will leave you able to set up but not recline, if my guess is right.