Then to clear the ballast tank you would open the air tank, the higher pressure air would then force out the lower pressure water (this is clearly why it can't be the same pressure as implied in your first comment), and then when you're done you have two tanks (your air and ballast tanks) both at 380 bar.
If that's how it works (which sounds better to me), you'd actually need a PV rated for 740 bar, which seems well outside our current materials science. Presumably you could increase the volume of the air tank relative to the ballast tank and therefore need less than 2x pressure, but the more you do that the more you're increasing your buoyancy, defeating the purpose. In fact (and I'm still mostly just spitballing here, not actually doing any math), I assume you really need some pressure greater than 2x, otherwise you're just cancelling out your ballast with your air tank. You need the volume of the air tank to be less than your ballast tank to actually end up with ballast tanks that make you sink without being totally counteracted by your purge tanks.
That would track with using this system in submersibles that are operating at ~300m and not for things like this which are trying to operate at 4000m. At 300m you could have a 400 bar COPV that is 7.5% the size of the ballast tanks, which sounds practical.
By filling the ballast tanks with 382 bar of air.
The human-interior of the sub is maintained at 1 bar to avoid compression demands (hyperoxic seizure triggers at approx 7.6 bar when breathing atmospheric gases; HPNS (High Pressure Nervous Syndrome) is common after 16 bar on a helium-oxygen mix), and decompression requirements.
Equalizing the inside-outside pressures would help a lot with the composite materials, and somewhat less so with the fleshy meat bags inside.