This appears to be a similar problem to one discussed
here in an earlier ElectronicsPoint thread. A small dedicated microprocessor would be able to sort your requirements and implement a single push-button switch solution.
There are many push-button switches available for use by the handicapped, some with extremely low force-actuation requirements. I have built some of these from scratch using optical interrupter sensors, which essentially require zero force, with the switch plunger restored to its unactuated position by a very weak compression spring. It all depends on how handicapped the end user is, whether or not they are capable of moving their hand or just a finger and what their range of finger motion is, to determine exactly what push-button switch to select. This must be your first design criterion: push-button switch selection.
It makes little or no difference at all what form the push-button switch contacts are: single pole, single throw, normally-open (type A) contact or single pole, double throw (type C) contact will work fine. Multiple poles are unnecessary. No motor, or actuator, or whatever large current will flow through the push-button switch. The switch is there only to inform the microprocessor of the user's intentions.
For safety, it is advisable to switch in a low-resistance load across the DC motor, actuator, or whatever load you are switching whenever the microprocessor commands the output to be OFF. Relays and contactors are too slow to perform this action. A full-wave bridge rectifier, connected across the load terminals with the bridge output connected to a MOSFET switch, will perform adequately without too much circuit complication. Or you can use a bipolar junction transistor (BJT). Either approach is satisfactory, but either one will require an optical isolator, controlled by the microprocessor, because the load is energized from a bi-polar source (the two 12 V batteries) whose polarity is subject to change.
Ordinary electromagnetic contactors can be used to switch the load current ON and OFF under control of the microprocessor. An intermediate relay or MOSFET switch may be required to actuate the coil of the contactor.
The ONLY function the microprocessor performs is a stateful inspection of the push-button switch to determine what the output should be: ON (forward), ON (reverse), OFF, or OFF with the load "shorted" by a low-value resistor when transitioning from ON to OFF. The load "shorted" condition only lasts for a few milliseconds after the user releases (de-actuates) the push-button switch. It is there only to dissipate whatever mechanical energy was stored in the motor, actuator, or whatever when that device was energized. You can eliminate this "safety" feature if you wish so the microprocessor has only to control two contactors: one for ON/OFF control and the other to select the direction.
I am confused by the images in your post #32. I see no motor there. Instead there is something that appears to be some sort of linear actuator with a small built-in motor rated at 24 V DC and 2.7 A. This is a far cry from your original speculation in post #24 of 12 V 10 - 15 A motor. What is going on here? Have you jumped off into the deep end of the pool trying to help someone who is handicapped and confined to a wheelchair? Are you aware that the linear actuator has only a ten percent duty cycle with TWO minutes ON followed by EIGHTEEN minutes OFF? Does the linear actuator have any safety protection to prevent it from being driven into mechanical limits and perhaps becoming "locked up" and perhaps requiring disassembly to make it functional again?
You can add such safety features to the microprocessor push-button switch control, and I would certainly recommend adding something to prevent exceeding the duty-cycle limitation. Replacing a burned out linear actuator doesn't sound like much fun, but a handicapped person may not realize when they exceed the maximum (two minute) ON time or wait the minimum (18 minute) OFF time before actuating the push-button switch again. Or perhaps this has already happened and is one of the reasons you are involved with this project.
Exactly what are your mechanical, electrical, and electronic experience and training? Are you a newbie at all this? Can you design and build a small microprocessor control for a push-button switch? It's pretty simple IMO but there is a learning curve and a bit of electronics assembly required.