Since this is an entire barrel, maybe your best bet would be to
pullout several bottles worth adding tartaric to each in various
amounts. (And top up.) The reason to add tartaric only is two fold,it
reduces pH more that malic or citric, and the excess is coming out if
chilled. When it does, it pulls potassium out too, further reducing
pH.

Your wine probably has very little tartaric acid, or is high in
potassium as a guess. I would think the later makes more sense.
Potassium Bitartrate forms when you chill wine, this reduces potassium
and tartaric acid. The colder, the less it can hold.

Personally, I might do this. I would pull a sample, add tartaric only
to a TA of 8g/l and remeasure pH. That would go in the refrigerator
for a few weeks. Then I would remeasure. Adding potassium bitartrate
allows you to pull this out at higher temps, you can add a few grams
to another sample to see if that works better. You want to get under
5C if possible, any fridge will do 6 C. Wine at 5 degrees C can hold
1/3 of the potassium bitartrate it can hold at 30 C, so you can still
pull a bunch out even at fridge temps. You will see a crust of
crystals form, you rack off of that while still cold.

Others may have better advice. This seems least risky, and you can
predict how much to add and what will probably happen this way. You
can do this anytime.

Regards,
Joe

If you can get it good and cold, you may want to add tartaric acid;

Trying to figure out how to drop the temperature that much. Also,
does it matter when (e.g., right before bottling or will anytime
work)? For instance, ambient temperature will be about mid-50's in
the winter. I could transfer the barrel contents to a stainless tank
and blast it with dry ice and get the temperature down. But I don't
know about getting it to 20F. Won't it, uh, freeze?

BTW, how much TA would fall out? If I acidified to 3.6 right , that
would put my TA at about 1.0 or so. Will 0.3 or more TA fall out?