if a contract is in place (max value), there will be no effect on his value at the deadline
as a rental it will also be equivalent (as his current rental value) or greater with teams knowing exactly where they are in the season and a rental market being established
I don't think there is an effect on his value between now and the deadline, zero reason for there to be. However - I think Ottawa would definitely like to pull the trigger prior to the season just because if we don't it likely makes for a drama filled few months.
His value is only what another team is willing to pay. The best of those offers. You don't have the option to keep him, realistically, so it's not like you're actually giving up a huge amount. Ottawa's management screwed them into this situation. What the other team is getting doesn't matter, because that's not what Ottawa's giving up. Ottawa can't trade one year/two months of a disgruntled Erik Karlsson and expect to get value for 8/9 years of a happy Erik Karlsson. Not unless they can drive a bidding war to that point.
His value gets reduced as teams begin to spend assets elsewhere. TB is probably going to do that any time. And when they do? They're out of the running on Karlsson both now and at the TDL. That drives his value down, because it's a major player out of the bidding war. They won't be the only one.
Teams don't like giving up roster pieces at the deadline. Ottawa's best chance to get real value is right now, with three (+?) teams in the bidding. They can gamble on some team going nuts for him at the TDL, but I doubt a contender is going to be willing to shake up their roster like that.