Are you ever going to run this "underwater, deep ocean rocket engine" above the surface for training, testing or or operations?
How are you actually intending to operate the thing? A deep-ocean (and water bottom change significantly!) operation would mean the outside of the tube is exposed to different water back-pressures, water temps (shallow, near-coastal down to your expected 4 degree bottom-of-the-trench worst/best case scenario.
Thus the diff pressure across the thing may have zero dp at one end (where water outside < gas pressure inside at the exhaust because the gas is leaving the chamber (but temp's are high inside the wall, very low outside the wall)) out to the other fixed end of the rocket chamber (where pressure inside >> pressure outside, but temp's are nearly the same.)
You're conditions and stresses are going to change down the tube. But, iff it is a one-time use item (like a thermite tube to weld underneath) who cares? You pay extra for the conservative wall needed to make sure it doesn't burn out the thermite liner, but even NASA couldn't tel how much their space shuttle booster shell was overdesigned. Turns out even that wassn't a exact, completely known value either.