I'd expect any coordination against the Triune's as too difficult between Rome and any German successor state.
What are people's thoughts on a Triune and Rome alliance? It sounds like a good fit if spheres of influence can be agreed upon.
I believe that the Triunes and Romans fit best as rivals.
Frankly it's borderline impossible. The Franco-Ottoman alliance is a false comparison. Long-term politics in this era has an 'enemy of my enemy is my friend' mentality to it. Alliances and enemies tend to leap frog. Countries threaten those that they border, because they can expand there, and ally with those who do not. If that country borders multiple states then you have natural allies. England bordered France and Scotland, so France and Scotland allied against England. In response England sought allies against France, finding them in Iberia be it Castile or more famously Portugal, but also in the Netherlands an Austria.
So now France has two enemies on its borders, so it looks at who borders Austria and allies with them. That would be the Polish, Hungarians, Prussians, and most famously the Ottomans. The Ottomans had a state they bordered called The Safavids who they got along with poorly, and so allied with the Persians' neighbours further east. This founded the the Mughal Empire's gunpowder skills under Babur, the first of the dynasty, via direct Ottoman involvement. Famously too an English group under Robert Shirley and his brother Thomas came to Persia to arm the Shah with modern military technology. Because the Ottomans were allied with the French to counter the Habsburgs and the English didn't like the French the English saw the Persians as natural allies to fight an ally of their enemy.
ITTL the same things have occurred and the same politics somewhat exist because Geography has not and will not change. The Triunes fill the geopolitical role of the English and are foes of Arles. Arles fills the geopolitical role of France and are foes with Hungary, who fill the role of the Habsburg Empire. Hungary is a foe of Rhomania, who fills the OTL Ottoman role. Rhomania is foes with TTL's Ottomans, who fill a Safavid role. The Ottomans in turn are foes with Vijayanagar, which fills (somewhat) the OTL Mughal role.
This is what I think like, basically.
However, I will put Triunia as a mix of England and France, perhaps the most potent and dangerous of all.
Historically, France's strength was its immense manpower in comparison to other European states, which bore fruit not only to provide soldiers but also in money via taxes, industry etc.
Meanwhile, England had its impressive naval capacity (France was no slouch here either though) and its mercantile powers centered around the City of London.
So, now England has an immense hammer to swing around to defend its mercantile interests (for example, by defeating and incorporating its closest competitor) and France has the English money it needs to march forth and secure its impressively long borders.
Arles, then, has no parallel, actually, because it's got precisely one foe at this point (Hungary is crushed, and besides, I have a feeling that they'll be looking more towards Germany and Vlachia than a Lombardy that will be protected by half of Europe for their own reasons), which is Triunia. Rhomania must protect Arles at all costs, to prevent the Triunes from gaining a Mediterranean coast, 'cause if Triunia manages to eat Arles, it could go wild in Italy, and theoretically repeat Napoleon's invasion of Egypt (that failed in OTL, yes, but that may not be repeated TTL).
Triunia is Rome's greatest enemy in Europe. Germany could have been held in check by Hungary and Poland and itself fight to protect Lotharingia as its meatshield against Triunia, but now that it is broken, all that is just... gone. Triunia could, in fifty years, really go wild in Germany and lil Lotharingia.
If it sets itself against Germany, the Romans are a logical ally, especially if the Hungarians find themselves pulled into the orbit of the HRE. No one would mistake such a relationship for one based on trust, but they are too far apart to be true enemies for long. Their interests simply don't conflict enough.
We know the Romans are entering a period when they pay less attention to Europe, but they still exist, and merely acting as an ally-in-being to the Triunes may allow both to deter Germany if and when it embraces revanchist politics.
If Germany is the target. But, now that Germany is a tad bit broken, Henri may well decide to add a new kingdom to his realm by invading Arles, which is the real threat to Henri, really. Germany has... just itself, and Lotharingia, and maybe Poland and Hungary, to fight with against the Triunes, while Arles has Spain, Aragon, Bern, Lombardy and potentially Sicily and Rhomania as well. The Rhine is a defensible barrier, but Arles is the dagger aimed at the Triune heart, and any Triune king worth his salt should know that.
Besides, simple revanchist pride: "Arles used to be part of France; let's reclaim it!"