So, heavily inspired by the "Luftwaffe Zeros?" thread, I'll propose another way to get to a worse/the same/better plane for the Luftwaffe.
If we go back to the specification for Germany's first fighter, Me-109 did not meet them, having much too high wing loading, but clearly outclassed those that did meet the specifications.
Lets suppose that the select process proceeds as OTL, but when Heinkel's is producing his rather competive HE-112B, he is without delay informed that the Me109 is selected.
This have not been without debate within the Luftwaffe. Me-109 is a high performer, but fragile, low ranged and unmaneuverable.
Luftwaffe decides it needs a long range, maneuverable air craft for dog fight Escort, broad based landing Wheels for front line service, CAP for the Graf Zeppelin etc.
He-112B goes into production for export as OTL, Heinkel makes his redesigned He-112C. Basically, being Heinkel he could modify He-112B or make an entirely new air craft , but the parameters is primarily long-range and maneuverability?
This insight kills the Me-110, but I guess FW could try their FW-187 proposal as well as it does perform on some of the parameters.
However, the main thing is the HE-112C. As it is intended as a quality aircraft to fligh off the Navy's expensive air fields (GZ) and it isnot intended as a high volume production, it does get access to the newest engines.
What would such a plane look like or perform like?
PS. Please dont derail it into a Göring or Udet would side-track this. The POD, whatever it is, specifies that the Luftwaffe wants such a plane in addition to Me-109.