It's a 10 amp circuit breaker by design, and the measured current draw on my 66 GT was about 5 amps, yielding a good safety factor of 2.0 for electrical fusing.
If your circuit breaker is tripping, I would immediately suspect a bad ground somewhere in your fog lamp system (including the rear running lights). A second area of concern would be a suspect switch itself.
The bad connection (if there is one) will most likely be where things are "getting hot".
It is just as likely an "Offshore" circuit breaker which is manufactured "cheaply"could be your cause of trouble. Also, make sure all of your "contact" (or "star") washers are all in place too. Poor contacts at the circuit breaker will cause it to overheat. I'm not sure what you might be re-doing wrong each time (or even if you are) but if you keep doing the same thing, the same way...you might just end up with the same level of insanity as many electricians have
.
In another thread (not sure who was having the same problem, possibly you?) a few months ago, it was suggested to use a "plastic separator" by another member UNDER the eye loops of the contact...I believe at that time, it may have been noted or suggested that this is "As the factory does". Possibly, the factory DID use those separators on certain applications but I am not so sure they did so on your 65 Mustang. If a separator is used, it
could add to poor conductivity.
Personally, If there is any sign of heat at the breaker posts (studs) when you get them down to look, I would start there. I would install another new breaker from a DIFFERENT source and with the "bare-naked breaker", first slip a star washer onto the studs, then the eye-connector, followed by a brand-new nut and assure yourself it is torqued down correctly. A quality circuit breaker should practically NEVER fail so my "guess" would be something like this is where your trouble is (agree with Randy, if the grounds are all good)