Is the third brake light in the tailgate? If so, I’d look for damaged wiring (from 14 years of flexing) where the wiring passes from the body into the tailgate.
It could be any of those things. Or it could be something else too. I did a quick lookup of what’s on that fuse, and it also handles ABS, cruise control, the cruse control laser (if equipped), TPMS, and a couple of other things. I’d start by checking the bulbs for shorts, and following @lion9car’s advice, but beyond that unless you’ve got some experience tracing electrical gremlins this might be best left to a pro.
I assume that the fuse blows when the brake pedal is depressed. Hopefully the problem occurs every time instead of being intermittent. This would make finding the problem much easier. By replacing the fuse with a brake light it will limit the current to the short to a safe level. Remove the brake lights and then see if the test light glows brightly when you press the brake. It most likely will still glow bright. You then disconnect suspect areas one at a time that the brake circuit supplies power to after the switch. Once the problem area is removed from the circuit the test light will dim.
There are some testers designed to help you find shorts you may want to purchase if you do this job yourself.
My Toyota Sienna 2005 had the same problem - the STOP fuse keeps blowing because of short circuits.
it started with Shift Lock problems, I had to push the manual release hold to shift gears
it also mess up with the rear top STOP light. As turn on head-light or frog-light, the rear top STOP will be turned on, but it is not on as step on break paddle
as long as turned on head light (drive at night), the shift lock problem were gone
when I figured out it was the STOP fuse blow out, I changed this fuse, but it keeps blowing again and again
How to fix it ?
Finally, I found out my driver side STOP light bulb was the source of all the problems.
This bulb was not burnt broken, but its supporting structure was burnt band to short circuits with the frog/side-light as shown in the attached 2 pictures.
After I changed this driver side STOP light bulb and changed a new STOP fuse, all the problems were fixed.