A tricky, high dexterity task with random 12G thrusts? Forget it.
The issue isn't so much the G LOAD, it's the random and rapidly changing nature of it. Hot welding slag, in a vacuum (as your hull was likely breached by the railguns) so not cooling much, randomly accelerating at 12g, or 115m/s/s, is probably going to be as lethal as a railgun round. A simple spanner ripped out of someone's hand would plausibly either penetrate the hull in a 12G manoeuvre, or come back and kill the engineer or obliterate the repair robot.
If your gun feed is damaged and needs welding, retreat for repairs. The close-in railgun fight at high G is a last resort when you're out of missiles and can't run away or they've caught up with you. I can't remember the exact quote from "The Expanse", but it was something to the effect of "CQB - we're dead. CQB? Oh that's close quarters battle. Railguns at close range. We never want to do this cause the chance of making it out is so low, basically its a fight to the death."
Ignoring all that - can this actually be done?
Perhaps - remove the "random" and you've got a chance.
With plausible near future tech I believe we could build a robot to do simple repairs under a steady high G load - the known, constant acceleration direction would factor into the repair. The robot could orientate the welder, itself, and protective panels in such a way that the slag is contained and no parts fall loose.
This is a huge AI / robotics project in itself even in the constant acceleration use case.
With such tech further refined, it may be possible for the flight computer to send a few milliseconds advance notice of its manoeuvres to all robots. This could allow the robot to either re-orient or secure its workspace and brace for acceleration before the rapid force is applied.
The downside of this is that your manoeuvrability is delayed by a about a hundred milliseconds. Your engines are unlikely to be able to spool up from 0 to 12gs in under 100ms so this may be a non-issue.
A network connection between the repair robot and the flight computer could also do a "synchronised dance" of the mutual interconnected tasks of a space fight and a repair. The ship dodges a salvo, then the robot gets in position and secures itself, then the ship starts a rotation, then the robot opens the outer panel and secures it, then the rotation stops, then the robot removes the damaged part, then then main engines fire, etc.
Certain G accelerations may be useful to aid in the repair - I've struggled to repair things and could certainly use a 12G force helping out. If the ship needs to side thrust any direction to dodge a salvo, and the actual direction doesn't matter, and if it moves left it'll help the robot pull out a stuck part, might as well thrust left.