We know how he acted in Aquitaine and Lombardy, and both of these events suggest a tendency towards diplomacy rather than warfare, as he never made any aggressive moves (except an alliance against the Pope and Charlemagne) during his reign.
We know that he didn't acted, mostly because it wouldn't have benefited him : contrary to Pepin/Carloman relationship, Charles/Carloman co-kingship doesn't seem to have worked out much, the latter not really interested on helping his brother onto was what essentially an inner matter (a move that likely prevented him to gather much of a powerbase among Franks, including on its own territory).
Doesn't mean he wasn't inclined to fight, but that one of the two brothers was one too much.
Friendship with the Pope was done by Martel mostly out of self interest, as he was quite happy to pull the wealth out of the churches in every other situation.
I don't think you have an accurate vision of the situation : first, churches in Francia weren't (at all) under pontifical jurisdiction but were the result of decades of nobiliar investement. Peppinids themselves supported the foundation of a lot of institutions, on which they trusted wealth (a bit like banks) that local clergy too often took as "gifts". Eventually, he certainlt didn't needed pontifical help to take care of it, as Merovingian did themselves in their time without the pope inspite of local clergy whining all the way.
Then, Peppinind interest on a relation with Rome was far from passing : since Peppin II (the father of Charles Martel), exchanges with Rome were essentially based on Austrasia becoming the western European hub on trade and political roads from the Mediterranean basin. Meaning that missionaries, for exemple (and you know how much politically relevant was mission when it comes to statue a political project on this era) gaining Peppinid support beneficied both Rome and Frankish interests.
To me it seems that Pippin III only used that friendship to 1. take the throne and 2. tell Lombardy to go to hell.
It ws certainly used this way, but certainly not the only one. After all, Charles never really had to take the throne, in spite of being a majordomo without king. That Peppin III had to take the throne (and to sacralise the entiere Carolingian family at the same time) may rather hints at an harder situation (mostly why Peppin had to throw a Merovingian one last time after his father's death)
There is no real reason why Francia should have chosen the Pope over Lombardy, it seems to have just been a convenient thing for the Karlings to do.
There's several reasons, actually.
First, Lombards were traditional allies of any contender or peripherical rival of Peppinids (see the informal alliances they tied with Bavaria, Aquitaine, etc.). Getting rid of Lombards made a lot of sense when it come to ensure inner peace (would it be only to make a point).
Then, a good deal of Peppinid legitimacy by the late VIIIth century was insured trough a political-religious policy : not only trough sheer legitimacy of their de facto rulership over Franks since decades, but also as bearers of an hegemonic project that was culturally heavily marked by the idea of chritian imperium. Whoever managed to appear as the protector of Christiendom (fighting raids, but also sending missions in Frankish Germania and beyond) certainly ensured its position.
Eventually, the takeover of Italy could be understood as well as part of the hegemonic project over the trade roads of western Europe : while Austrasia was at the core of what remained of Mediterranean/North Sea contacts, most of the gold (coinage mostly, but not just that) that got carried in Francia came from Byzantium. This gold had an important symbolical and political value, especially with the establishment of carolingian feudalism and the widespread wealth redistribution it implied.
Of course, one could find other reasons, but I think these were really too important to be ignored.
Frisia held a scrap of independent land on the North Sea coast, and even Charlemagne just ignored it until it actually struck against him (in Widukind's rebellion).
Independent would be stretching it : at this point, Franks controlled most of the wealthier part of Frisia (Rhine's mouth, frisian harbours), and whatever remained was definitely crushed, without known sole ruler since Poppo. Basically, since the mid-VIIIth century, Frisia ceased to exist as a political player.
Saxony had spent a long time with a border with the Franks (going back to the 6th or 7th century), and most wars had been for tribute rather than conquest.
I don't really see the relevance to oppose wars of tribute/submission and conquest : as it happened for Frisia (regularly targeted by Franks since the VIIth century), the Frankish campaigns first imposed some sort of subserviance, admittedly often rejected, and eventually turned in full-fledged wars of conquests. It can be said about Frisia, Aquitaine, most of patrician Provence, Bavaria, and it was what happened to Saxony.
There is no reason why that could not continue in an ATL.
Actually, quite good ones.
First, Saxons opposed any idea of Frankish being able to set up a dominating presence in the region : whatever tributes, missionaries, etc. The idea din't thrilled them. Lo and behold, at the contrary Franks were pretty much about putting Saxony under their thumb.
Then, you had a more vested interest of Peppinid rulers to deal with Saxons more decisively : until the VIIth century, the commercial/economic and generally productive centers of Francia were more centered around Seine and Meuse basins; which began to change with the decline of Mediterranean trade due to Romano-Persian and afterwards Romano-Arab wars. The main hubs indeed shifted more decisively in Austrasia, where Peppinid power base was centered (and not unrelated to their political rise). Saxon raids became politically and economically far less tolerable, and when Saxons decided to raid Thuringia for shit and giggles, it was the last drop : Frankish Germania was the "new border" of Peppinids, and they didn't wanted anyone to mess with it.
Most of Charlemagne's conquests were because of convenient circumstances, which would be less likely to occur without Charlemagne there:
Every, litterally, every place you mentioned already knew Peppinid interventions up to campaigns before Charlemagne : Franks didn't suddenly expanded for shit and giggles, but because it was set on a longer geopolitical continuity.
Lombardy was invaded because Frankish campaigns in Italy didn't stopped Lombards to ally with Peppinid's rivals or contenders.
Bavaria served as Peppinid target practice since Charles Martel
Saxony was a hub of raiders regularily fought against by Merovingians, and Peppinids had even more interests in Austrasia and Frankish Germania to not let Saxons have their way
Avars/Slavic border was a matter of intervention of Franks since centuries : while wealth and slave acquisition certainly played a big role there (and you notice that it wasn't one of the main Carolingian campaigns), Frankish presence in the region is attested at least since the establishment of Samo's kingdom.