Once you remove the Becker, bench test it w a 12 V power supply and two loose car speakers.
Find out what gets warm and hot.
If you felt the heat while driving, could it have been heat up from the trans?
My 300d was like that. I put in a piece of insulation to heat shield it.

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1979 300D 220 K miles
1995 C280 109 K miles
1992 Cadillac Eldorado Touring Coupe 57K miles SOLD
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1979 240D 140Kmiles (bought for parents) *SOLD.
SAN FRANCISCO/(*San Diego)
1989 300SE 148 K miles *SOLD

That's not a bad idea at all.

The car was not even running when I felt the radio get hot.

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1991 F250 super-cab 7.3 IDI. (rebuilt by me) Banks Sidewinder turbo, hydroboost brakes, new IP and injectors.
2003 S430 - 107K
1983 300SD - Tanoshii - mostly restored ~400K+.
1983 300SD - Good interior. Engine finally tamed ~250K.
Monark Nozzle Install Video - http://tinyurl.com/ptd2tge