You have a relay and a solenoid that must be addressed. Relay would be easy to change to a 12V unit. Solenoid is going to be an issue that can be dealt with by resistors. Not an expert here but someone with knowledge can compute the resistor value based upon the amp draw and ohms resistance within the solenoid. Contacts in the governor and the carb switch (kickdown switch) should not be an issue. Here is a link that should help you understand the electrical needs you have.
Solenoids are rated by the amount of push or pull that they can create. Just for giggles, see if you can determine how much effort it takes to actuate the load being driven by the solenoid. A gun trigger pull scale might do the test. The length of solenoid shaft movement is very important, too. Solenoids will have many shaft travel specs. Check into what you need in both respects.
Good luck on finding a solenoid. Built into transmission side. (see picture on page two of the manual) and here is an ebay picture. 12V pushes with twice the power of 6V. a resistor with sufficient amperage capacity slows it down enough that it comes out 6V. It was a common thing to do when 12V came out and you could buy resistors to run 12v into 6V radios. Just a slightly different application.
All of us replying assume your car has the semi-automatic transmission. That one does have the governor and solenoid. If you have a simple
three speed manual transmission with fluid drive there is nothing on the trans. to convert. My shop manual for that period of Mopars
with Tip-Toe, Gyromatic etc. specifies 20 lb force by the solenoid when energized with 6 volts. Chrysler changed to 12 volt negative
ground for the 1956 model year and discontinued the semi-automatic transmission in favor of fully automatic Power-Flite.
Therefore there were no factory 12 volt solenoids. A specialist on old transmissions has told me he was able to find a suitable
resistor to use on a car he worked on. I don't remember whether reversing the polarity affects the solenoid. It would be an easy
test to make with the solenoid in hand and a 6 volt battery. Rewinding the solenoid might be possible but not easy to open the case. GCW