'65 Dart No brake lights. Brake light switch failure?

Correct. The brake lamp wire doesn't route thru the turn signal switch in 1965. It did by the time of my 1969 Dart. You can tell because the replaceable plastic cam in later cars has 2 wires on it vs just-plastic in earlier ones. Check rockauto photos.

All the switch does is connects 12 V (always hot input) to the output when the stem is extended. You might remove it and check with an ohmmeter, and insure that neither terminal connects to the bracket (ground) in any position. To go further, the body connector is behind the left kick panel, where you can verify wiring to it and from there to the rear bulb. You can find wiring diagrams linked here. The heat suggests a short to ground right within the switch, but could also be a short circuit anywhere on the downstream wiring. Strange that a short wouldn't blow the fuse.

BTW, adjust the switch position so that it senses brake pedal motion, but doesn't prevent it. The pedal must not bottom-out on the switch. If it does, that might prevent the Master Cylinder from fully retracting which can cause brake lockup (vent port isn't exposed). But, the switch should be close enough that it is assuredly open with pedal up. If not, your battery will run down overnight from the lamp staying on since supply is always hot.