> Do you think a ground plane will provide a suitable heatsink
> for the IC?

Adding heatsinking would be no bad thing

There are clip-on and solderable IC heatsinks available. You might
find one on a monitor/CRT board. Or make your own with a bit of
'tin', eg an actual tin can or screening metal found in many appliances.
Easy to work with, solderable, free
--=20
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