TH,
If it were my car, I would make a slide bracket for the two sides of the radio.
Think something along the lines of what furniture draws use.
The brackets may/might be attached via JB Weld to the inside side walls of the consul. Spacers may be needed for distance to the sides of the radio.
The radio would be bolted to the face plate as you have it now. The face plate will not be bolted to the consul, but will now move forward with the radio when you want to pull it out/forward so as to have access to the transmission dip stick. (This of course will require some cutting of the front of the consul behind the radio face plate). All the cutting will be hidden by the sliding face plate.
The wires would be long enough so as not to have to unhook them.
The distance the radio pulls out forward before it comes into contact with the stick shift should be enough to have access to the dip stick.
After checking the dip stick, push/slide the radio back into the consul face.
Keeping the radio locked in place(from behind)will require making some sort of mechanism that will be easy to use and sturdy. Maybe something like a pin that goes through one or both of the slide track rails?
Others may chime in about how much time I have on my hands to come up with such silly ideas.
Cheers, Roger