Why doesn't respond well in reverse?

No not the whole transmission. Could be a band or servo. It still has to come out and be tore down to fix.

Shifter cable/ linkage needs to be adjusted.

that is a hum dinger, largest ever posted, in one shot.
very good.!
one word , the transmission slips. (never drive a slipping tranny)
overheat indicator.! that is key.
many newer cars will change the transmission shifting any time
the engine over heats.
and overheat wins over transmission , every time.
if the engine burns up, what good is any transmission.? nada.

solve overheat first.

slip or flare.
is when the engine RPM rises fast and the car dont go faster.
SLIPS.
and slipping burns up trannyclutchs and increases the damage. inside the transmission, dont do that. dont allow it,
park it and fix it.

on some tranny there is a temperature sensor that the TCM
watches and changes shift patterns based on AFT fluid temps.
and if the ATF fluid gets too hot , it takes actions. to keep the tranny from harm.

why not get a scan too and see what DTC errors there are..
DTC, engine overheate.
DTC< tranny overheat (caused by above)

try to adjust clutch cable until the reverse makes no rattling sound,

its not reading the speedo signal properly, does you speedo wave about sometimes ?

sounds like the lincage bolts loose giving extra movement

your problem lies in the internal parts of your transmission which are the clutch linings, they are already worn out and starts to slide.

sound like your your trany is on it's way out but you can try changing the il and filter may give you some time until you can get it looked at

I would have to say that your transmission failed. The Tranmission was trying to lock up and the engine was being forced to labor extra hard and that is what more than likely created the extra heat.