But they can't sign Davis with $7.6 million in cap room. That's why they have to renounce to the MLE. The MLE is always there as a cap hold.
You're not getting it, and I can't explain it any better.
The team loses its MLE as soon as it has cap room above the MLE-hold (and the LLE- and traded-player exception holds). Once that MLE is lost, there's nothing for the team to renounce. They don't have it, so they can't renounce it.
So, how do they sign Davis if you're right? What is "lost" is the ability to use the MLE to sign FA, it keeps counting as a cap hold. Hence, you have to renounce it to make room.
They also have a Mid-Level exception for $5 million and a Traded Player exception for $5.5 million. Even though their salaries put them $6.5 million under the cap, their exceptions are added to their salaries, putting them at $53.5 million, or $4 million over the cap. So they actually have no cap room to sign free agents, and must instead use their exceptions.
Teams have the option of renouncing their exceptions in order to claim the cap room. So in the example above, if the team renounced their Traded Player and Mid-Level exceptions, then the $10.5 million is taken off their team salary, which then totals $43 million, leaving them with $6.5 million of cap room which can then be used to sign free agent(s).
You're disregarding this part:
The Disabled Player, Bi-Annual, Mid-Level and Traded Player exceptions may be lost entirely, or the team may never receive them to begin with. This happens when their team salary is so low that when the exceptions are added to the team salary, the sum is still below the salary cap. If the team salary is below this level when the exception arises, then the team doesn't get the exception. If the team salary ever drops below this level during the year, then any exceptions they had are lost.
Once this happens, the MLE disappears, and obviously, the MLE-hold disappears, as well.
AS coon says, "the team doesn't get the exception". Not "the team has to renounce its exception". It doesn't get it at all (or has it taken away) and thus doesn't have to renounce it.
We're basically talking semantics here, because either way the team doesn't get to use the MLE. However, you're incorrect as to why. The situation you're talking about is when a team has, say, $7 million in "cap room", but can't use it because there's an MLE hold. In that case, you renounce the MLE (and your LLE), and you can sign a free agent to a $7 million contract; if you hadn't renounced the MLE, you wouldn't be able to.