if i may ask, what brand/model the amp is ?
you may be better off just tying the red/black wires together and using the amp STBY/OPR switch or the power switch when you want to use it assuming its not being used for two way communications, depending on what type of amp it is and whether its solid state or valves.
edit:
its probably this amp ?
https://www.amazon.com/Amplifier-FT-817-IC-703-Enhancing-Transmission/dp/B00SMH5P8Uif so, you could get away with tying the small red/black wires together and just use the on/off switch to activate it, provided you memorized the band switch dial or physically marked it with a small black marker so you can change bands with the power switch off.
in any case, you will probably be able to do about 20-25w max on AM @ 13.8vdc, anything higher may introduce carrier saturation and envelope distortion on AM.
for cooling, i suggest a 12vdc computer case cooling fan placed on the top of it with the air flow going towards the heat sink fins.
tip: its getting too hot if a crayon starts melting within 5 seconds (easiest method).
good luck !