Thermal pads are more or less crap, I tried a lot of different ones.
best thermal coupling is big copper plate with paste.
The main thermal copper plate (your heat sink) is grounded to the computer battery minus pole as is the metal case. So there is no danger in using a big copper shimmy to connect the main heat sink with the metal back plate.
If you want to be sure, use an ohm meter and measure battery minus with heat sink and with back plate.
Your problem might be a not plain fitting processor / heat sink connection with not enough pressure.
For best effect, remove all, clean surface of cpu and copper heat sink add a copper shimmy with paste on both sides in between cpu and heat sinkĀ and one large copper shimmy (or as I did a few small ones) direct on top of copper cheat sink were the processor sits underneath. You want to have a bit pressure from the bottom plate on to the assembly so you have the best heat transfer connection.
Read the last entries of my diary below and you get an idea.
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1981 soldered my first Sinclair computer 1K, tapedeck * 1984 build and sold IBM clones 8Mhz, 512K, 20MB HDD * 2018 messing with ultrabooks