I'm not sure that increased trade in the North Sea leads to the Carolingians by default. Also, I'm going to have to contest your claim that the Islamic conquest of the Mediterranean was responsible for increased trade in the North Sea. This is Pirenne's argument in "Mohammed and Charlemagne," isn't it? I recognize it by smell. It's kind of dusty and reminiscent of 1929.
First my point isn"t that North Sea lead to Carolingians, rather than without Islamic conquests, and achievement of Late Antiquity trade continuum, Franks could have likely tried to obtain oriental goods and gold by Mediterranean flux rather than Baltic (who wouldn't be reached that easily by an equivalent of Arabic/Baltic long-range trade).
I never said Muslim takeover led to the appearance of North Sea trade while it certainly helped to make it growth : the presence of trade directed to Arabo-Muslim economical continuum definitly helped it and the presence of an North Europe/Mediterranean trade is well attested and the influence of Islamic coinage there as well (imitation of dinar, and close relationship with carolingian denarius)
My point is without a Muslim takeover of Mediterranean basin, the North Sea trade would have probably not that easily taken by Franks, and representated less of a focus.
While Pirenne assumptions were proven wrong, the intuition of the link between Islamic takeover of Mediterranea and Frankish growth was good, and still used in a "post-pirennan" consensus.
Pirenne held that the economic situation in Europe and the Mediterranean was largely unchanged during the collapse of the Roman Empire
Rather than unchanged, we can still argue that Mediterranean trade was in the same continuity during all the Late Antiquity. While McCormic argues that it was already declining (basing himself on non-economical sources), it was still enough to provides Franks and Visigoths oriental production and gold, something that really declined with the VII century (and, if you allow me this point, probably helped the conflicts that popped up then, leading to unification of Francia and "gothic disease" in Spain).
Basically, Pirenne's argument doesn't work and I'm not willing to buy the idea that the Carolingians were triggered by Islam's conquest of the Mediterranean
It basically is, nevertheless : without the appearance of a Muslim takeover of southern and western mediterranean basin (with the destruction of the only worth of mention fleet in the region during the capture of Byzantine North Africa and Spain), favouring a long-range trade passing no longer trough Rhine (favouring the passing of a sliver-based trade) or Provence loosing its intermediary position that it acquired in the VII century, trade flux passing from Italy, and Torcello.
Richard Hodges links that with a shortage of gold and decay of Byzantine economical continuum, but estimates that the regional trade must have helped to regain a balance.
"One in no way contigent upon the Arabs in the sense Pirenne suggested, but instead a consequence of Byzantine military and economical failure".
That, I think, is the keypoint : while Pirenne assumption was proven wrong, the core of its argument : such as the Islamic conquest created a new trade network in Mediterranea, leading to compensate (rather than create as Pirenne originally tought) the growing North Sea trade (the conquest of Frisia and Saxony being part of it)
That the issues of mediterranean trade, as underlined by McCormik, began earlier than Islamic conquest is possible but it didn't break the old trade networks and flux outright. A double Arabo-Islamic flux (one from Al-Andalus, and one from Abassids) did that, and helped the growth of Northern Europe trade (something that backfired, eventually).
I think that without Islamic takeover, and the maintain of a slowly declining Byzantine economical continuum, you could still end with a Frankish dynamism more focused on Mediterranea due to the absence of a need of baltic trade for recieving oriental exportation goods. I would say more focused on Italy and Bavaria/Venice/Central Europe (OTL, the Rhine/Alps/Venice flux was enough important for Carolingians to try taking the region and while failing at, having Venetians coining Carolingian denier)
Of course, we can't explain Carolingia expansion only by *one* perfect thesis, a frankish takeover of continental North Sea would only be more delayed and limited rather than butterflied.
I would eventually point the importance of the fight against Arabs in the Peppinid/Carolingian political takeover of Frankish kingship : without them, no easy pretext and situation for conquering Aquitaine and Provence (that acted more or less independently then and were weakened by Muslims raids), no good "ideological" fundation of their prevalance over Merovingians.
It won't be enough all by itself, but the peppinid struggle for hegemony would take longer without Islam.
Which likely means, back to the OP, more delayed christianisation of Northern Europe.