Well, what about to take a different way to think  and
to pay attention to the method you make the coffee.
Using French Press method, at least in my case all
the time "enjoying" the coffee, at the end, I have small
rest of coffee in my cup. If this happens to you also,
then I guess your problem is, that with the fact that
the coffee is hot entering in your Thermos and that
you did not extract completely the coffee powder,
obviously the hot water and the coffee powder rest
will continue have a chemical reaction, while being
in your Thermos, Therefore I suggest that while
putting your coffee from the French Press into your
Thermos you should use a filter ( maybe paper) to
guarantee to extract the rest - quantity of coffee powder
out of your coffee.
(maybe you should use a more rough grind and then
you could even use a filter of cloth etc... up to extract
all rest of coffee you might have inside)
RESUME : I think your problem is rest of coffee powder,
that due to very hot water continues with it's chemical
reaction.
Hope this helps,
saludos from spain
linus