Well I guess if there are no survivors, you can claim salvage, but:
1) When you go so far as to ensure there are no survivors, it isn't self-defense anymore, and you'll have to prove you didn't just do it to claim the ship.
2) They can expect a reward from the ship's rightful owner - that's what salvage means - but they can only have the ship itself if no rightful owner shows up.
3) If the pirate captain in question was the rightful owner (i.e. the ship wasn't stolen) then it's probably going to be confiscated by the authorities.
Alright, think I now know what I'll be doing.
"When you go ..."; wait,
WHAT???
They killed the wounded? They wouldn't accept a surrender?
OOOOOOOOOOHHHH! That chaps my lawful good heinie! Somebody needs a spankin'!
Besides, it's like throwing credits into the abyss. Live prisoners are worth much more, at least in my TU. The Imperial authorities can pump them for information about their associates and fences.
You could try to claim salvage, but you would have to drag the ship to the local admiralty board who would make the decision what to do with the ship (and/if they awarded it to the PC's legally). However the board would probably ask some pointed questions as to how the ship came into the PC's grubby hands in the first place...
Eg:
Mr PC, this pirate has FGMP burns - a restricted wepaon...can you explain this?
Mr PC, the local police would like to ask some questions about this vessel you claim as war booty, their records some some irregularities.
Mr PC, we need your transponder logs to investigate events, dont worry I'm sure you have nothing to hide.
Mr PC, your lawyer has said he wont continue to argue your case until you pay him a 50,000Cr retainer.
Mr PC, the board will get to your case soon, we expect it to be resolved in 6-9 months.
Couple of points.
An altercation between ships in space is under the jurisdiction of Imperial law, not the locals.
I'm not aware of any Imperial restrictions on personal weapon ownership on ships in deep space - only who you point them at.
The Imperium's been dealing with civilian ship-vs-ship altercations for over a millenium. I suspect by now they've figured out a way to resolve the case in a manner that doesn't encourage witnesses to wander off or encourage affected parties to cut the authorities out of the loop - especially with TL15 tools and computers available for forensic analysis. The longest wait I see is maybe a month, if you happen to be somewhere where they have to fetch authorities and equipment in from a "real" starport to conduct the inquest.
As to the later idea that, after a thousand years and more of civilians pot-shotting each other under Imperial jurisdiction, the Imperium doesn't have the tools to analyze the battle damage and recreate what happened with fair accuracy is just silly. A thousand years of it, they'd come up with
something. If nothing else, they'd require a couple or three black boxes scattered aboard to capture sensor data and ship operational data at the instant a weapon was fired and at the instant any ship system flashed a damage alert. That'd also give them valuable clues if a ship suffered a solitary mishap. Woe betide you should those boxes be tampered with, missing, or all suffer damage that erased their data in the same combat.
A ship that comes under fire is entitled to defend itself and, given potential police response times of as much as two weeks, is entitled to destroy its attacker if able. Such an act is a benefit to the Imperium as a whole. To imagine a system in which authorities are incompetent to tell attacker from defender, and with perverse penalties for self-defense in an empire where help may be completely unavailable to the defender, is to actively discourage spacefaring and interstellar trade. I don't see the Imperium allowing such a state of affairs to go on for 1100 and more years, nor do I see such a state of affairs creating the kind of setting that Traveller presents. You might have a few odd pockets of bureaucratic incompetence here and there, but the norm will be to settle it quickly and to the satisfaction of the innocent party.