"... Berríos is a middle of the rotation arm (low-2/high-3) who likely wants to be paid like a frontline guy"
Aye, there's the rub! There are three possible outcomes here:
Berríos improves to the point where the Twins can accommodate him (possible, but not likely)
Berríos falls apart (less likely)
Berríos' performance gradually falls off as he ages out of the League (most likely)
Assuming scenario 3, considering a trade makes sense. The next problem is not what the team could get, but what the team needs. Surveying the positions, the question is, who is/could be on the market who represents an improvement?
A 1 - 1 trade for Berríos would probably not deliver bigger talent at catcher, corner or middle infield than what the team has now
Outfield? The need for a bigger bat and better range is obvious, but which team will be selling a proven veteran that the Twins can swap into their assemblage of journeymen and walking wounded? Any names come to mind?
Pitching? Clearly the need is greatest here; but this is also the tightest market.
A true closer who instills fear and doubt in the opposing lineup cannot be had for Berríos alone
A journeyman long relief horse is more likely
Or, if management decides the team can no longer contend this season, do they bargain for a busload of prospects and sort through them for the next 3 - 5 years before we know if the deal was worth it?
If Berríos alone were not enough in exchange for the talent the team wants, could Sano be made part of the package?
Management has assuredly already asked all these questions. Assuming the Twins are pulling out of their tailspin, if Berríos is traded for a reliever before mid-June we'll know that management is sticking to it's guns. If he's still part of the club in July, then they either believe he's a piece of of a post-season drive or they still think they have negotiating leverage.