I have also looked for opportunities to use those nice Barons Wars miniatures coming my way. And the Thronstreit looks like really great theme for them (I had the Albigensian Crusade in my mind because Peter of Vaux-de-Cernay's History and the Chronicle of William of Puylaurens are easily accessible and really packed with action filled scenarios and detailed force descriptions).
I quickly skimmed over some literature (you can click these titles but unfortunatley they are all in german, maybe you can use the bibliography in Mamsch).
Winkelmann, E.: Philipp von Schwaben und Otto IV. von Braunschweig. Leipzig 1873.Abel, O.: König Philipp der Hohenstaufe : mit ungedruckten Quellen. Berlin 1852.Mamsch, S.: Kommunikation in der Krise. Könige und Fürsten im deutschen Thronstreit (1198-1218). Leipzig 2009. - this one seems especially promising for wargaming because it dedicates the whole chapter 5 to the communications during conflict. )
It seems like there where a lot of "minor" battles and fighting going on. "Minor" because there would be the full force of various dukes, marcher lords, counts and kings be involved (the older literatures gives 40.000 for Ottos army in 1203, a size Philipp could not engage according to the literature).
This one could be a really great project. Stefanie Mamsch has kindly included a timeline (of which I picked a few parts I think would be interesting for wargaming).
1202 November/December campaign against and murder of the bishop of Würzburg (chancellor of Philipp, gets murdered in his own city while going to church in the evening - great infiltration scenario in a dark city) by his own Ministeriales the Ravensburger because there where ongoing feuds between them
1203 there are mentioned devastations of Thuringia
1204 March/April a looming battle that is prevented because count palantine Henry changes side to king Philipp
1204 Philipp campaigns in Thuringia again, then in September the landgrave of Thuringia and the king of Bohemia are subjected by king Philipp
1205 Cologne is sieged, aborted after 5 days and campaigns against Neuss, Aachen and Bonn follow
1206 July second campaign against Cologne and battle of Wassenberg, 1207 March some messengers to Rome are intercepted by Philipps men
1207 a peace treaty until 24.06.1208 is signed
21.06.1208 Philipp gets murdered by Otto VIII. of Wittelsbach
24.06.1208 end of the peace treaty Philipps forces are already in Bamberg, an awful lot of lords prepare for war, but no big campaigns take of, negotiations start
1209 March Henry of Kalden the imperial marshal fights the king slayer and throws his head into the Danube (awesome battle on a bridge scenario between to mighty warriors - maybe a bit too romantic).
So we have some big battles, some smaller devastations and some skirmish scenarios. One probably has to go back to the sources as modern german historians tends to be a bit lighter on the battle describing side of things.