Thank you thank you thank you! Docker processes randomly exiting with "137" suddenly make a lot more sense!
137 - 128 == 9
... that's SIGKILL, probably via some sort of pipe/subshell. We'd figured it was OOM-killer or something similar, but now it seems like there's a bit more of the dots connected between.
Who can/will send random `kill -9`s? Probably the kernel, or some sort of supervisory process.