The first aspect we should make clear is that Denmark was seen as a greeater power than Sweden at the time of the intervention. But Sweden had made the military reforms which put them ahead of Denmark in OTL. Denmark had some weaknesses which made the intervention harder for Denmark. First was inferior military command, Christian IV was an average military commander at best, while other commanders was raised among the small Danish and Holstein nobility, the Danish army was build on a small high quality royal guard, a high quality marine force, urban defensive militias and small weak peasant militia. In time of war, the Danish army used foreign mercenaries (The navy on the other hand was one of the better one in Europe). This structure meant that Denmark needed large amount of currency reserve to go to war. The defeat in OTL meant that those currency reserves was depleted.
Sweden on the other hand had superior military command based on the Swedish burghers and nobility and a conscript army. This meant war was cheap for the Swedes.
So did Denmark have some strengths versus the Swedes. The answer are yes, Sweden was fundamental a rogue state, it often broke agrements, it didn't declare war before it attacked and because war was cheap and gave giant fortunes to the Swedish elite, it had a interest in keeping wars going. The Danish royal house was old with a high degree of legitimacy and it was better than Sweden at diplomacy.
But what does this means? It means that if Denmark won at Lutter (it was closely run), and avoided being defeated, they could likely hava made the emperor meet tham at the negotiation board.
So what demand would Denmark likely push. The Danish king would demand the emperors respected the Lutherans right to practice their faith, he would likely push that the emperors accepted the Danish king as protector of the True Faith (fundamental making a attack on Protestants a casus belli), the right to raise customs on the Elbe (and maybe the Oder and Weser), Bremen-Verden became a hereditable fief of his son Frederik and Schwerin a fief of his son Ulrik. Beside that I expect that he would demand that colonel of the Lower and Upper Saxon Circles became a hereditable title for his line and that Pomerania, Pinneberg and Oldenburg in case of their line dying out would go the Christian or his descendants.
Christian IV would on his side accept the Emperor as king as Bohemia (which would keep Lutheranism/Hussism as state religion) and they would likely reach some kind of agreement about Palatinate (as Frederick V was married to Christian's niece, I expect that Frederick would at least keep his Rhineland possessions).