According to insiders, last summer there weren't teams willing to go to the Doughty term/dollar with Karlsson in free agency. With the Sharks going 8 x 11.5, they would probably have to retain to move him.
The only way I can see him getting moved is if Doug Wilson is canned, and the new GM wants to rebuild. Otherwise, what incentive does San Jose have to move Karlsson? They clearly believe in him based on giving him that contract a few months ago. Why would one season where everything is going badly in San Jose convince them that they made the wrong choice?
Without retention, Karlsson would not go for very much right now. That's because there would be so few teams prepared to take him. In this scenario where San Jose wants to start completely fresh, something around Karlsson to Detroit at the draft might make a ton of sense. If I recall, Detroit was one of the teams that was always rumored to be an ideal destination for Karlsson. Maybe that has changed now that they don't have the legendary Swedish core and they are in a rebuild. If Yzerman was the one pushing for Karlsson in Tampa, maybe he'd be interested to bring him into Detroit to build around along with whoever they get at 4th or better in the draft.
If Karlsson is moved at full salary, it would be for a really soft return because so few teams can fit him in at full cap, so few teams will want him at full cap, he has a NTC, and if San Jose moves him knowing all of this it shows that just like with Nashville and Subban - San Jose would be motivated to wash their hands of the player. The team acquiring him would be making a great gamble because a healthy motivated Karlsson in the right coaching system could be a franchise player well worth the 11.5M cap hit.
To San Jose:
2020 32nd Overall
2020 2nd Round Pick
DET 3rd or 4th Best Prospect
In the extremely unlikely scenario that San Jose wants to wash their hands of Karlsson without retaining, and Detroit wants to push this soon to undercut the rebuild, I could see a trade like that. I highly doubt San Jose in any way wants to move on from Karlsson though. If you're Karlsson, it wouldn't be productive to ask for a trade because it's highly unlikely that anything other than a rebuilding team like Detroit or funnily enough Ottawa could fit him in.