Why does it have to be right away?
You listed it among mentions of Umayyads and Charlemagne : it looked like you supposed it would happen in the same time.
OTL, Iberian Christian mariners only began to sail along Africa after 1250 or so, 500 years after the PoD. If (due to Christian control of Andalusia and southern Portugal), that begins in 1000 instead, surely that is a reasonable knock-on, and a considerable change.
The changes for what matter spanish structure and cultures, particularily on seafare, would be great enough for that speculating on what happen 500 years later is a bit moot : we could say, with as much confidence, that an iberic seafare really appears after its IOTL counterpart.
Simply said, there's no cause to effect relation with the PoD on that matter.
Why does it have to be in the 8th century?
Because, by the VIIIth century, a simple comparison betwen Gothia and Francia highlight wells who's on the rise and who's not.
Again, we can speculate for what happens 200 or 500 years after the PoD, but it's a bit irrelevant giving it couldgo several ways with relatively equal chance.
OTL, there was no significant Spanish state until 1250, and France towered over Western Europe. A Spanish state which forms in say 900 makes a big difference.
Gothic Spain was a spanish state for what that matters, as well Muslim Spain. You're confusing a spanish state with the same background than ITOL, with a Spanish state forming differently.
How long must that weakness persist?
It's hard to say : at best you could have something of a shock therapy after the likelt Frankish overlordship or at least domination. But to be honest, the decentralized and anti-dynastical structure of Gothia was really well entranched and the various takover never really managed to pull it off historically : not to mean it couldn't be done, but as much you could have a unifying reaction to Frankish dominance, as much you could see peripherical regions of Spain (Galicia and Lusitania, for exemple) establishing principalties of their own with a split between various Spanish entities.
Goths will have to deal with a relatively long decline (one century at best IMO), from which they will emerge anew or shatter in pieces. Or both (you may end with a quite weird equivalent of late carolingian feudalisation, especially if eastern Spain fall into Aquitain/Frankish influence).
The Moorish conquest of Iberia killed off Iberian nationhood for 500 years
I don't think it's remotely true. The idea of "nation" as we understand it nowadays (as in a more or less free association of humans groups sharing similar values and considering themselves as a whole fused with institutions) didn't exist : how the conquest of Gothia was merade possible in first place was that various potentes simply felt little to no solidarity with each other (which was aggravated, admittedly, by the ongoing civil war).
Revolts such as Paulus' in eastern Gothia, with the revendication of establishing a separate kingdom; or the fact that several provinces (especially Gaul's) were xempted from decrets highlight that you had no such thing as a nationhood there.
Heck : how several Goths readily joined up with Arabo-Berbers does point that we can't talk about nationhood in this context.
What you had was the fusion of ethnic identities, as it happened in Gaul or Italy, under the "Gothic" label : but how it quickly disappeared both in al-Andalus and various Christian principalties shows its limits.
is it certain that Gothic political infighting would last as long?
It was something deeply ingrained since centuries, and litterally part of the political culture and institutions of Gothia. It was less infighting itself, than that fearing the establishment of a dynasty (or that a yahoo like Roderic could impose itself on them), that could impair their power, managed to provoke a cycle of anti-dynastic succession whom a good part
did ended as avoiding civil war.
Simply said : it was part of their zeitgast, their cultural and political conception.
Now, I entierly agree it couldn't last long if it was harming their interests. I even think that it would eventually evolve out of a new situation as it happened in al-Andalus historically.
But as said above, I do think it would certainly come from an outer incident (like it happened with Frankish takeover of Italy or Saxony), mixing up ITTL Frankish and Gothic features.
The best case scenario would be Carolingians pullling a Theodoric and being content with establishing Gothia as a client/protectorate while having only limited conquests : and let's be honest, it's not really unlikely, especially in the case of an Arabo-Islamic threat popping up earlier in Italy. With the caveat that with Carolingian decline, you might end with return to old ways; but the early feudal model could be likely imported in at least part of Gothia.
If not, I'd be leaning to envision a factual division of the peninsula along the traditional line between the eastern third and the rest, at least for a time. The worst case scenario being that the remaining 2/3 explode in several principalties. At this point, I do not think that Frankish or Franco-Aquitain dominance (which would be barely felt in some regions IMO) would last forever (altough I could see an enlarged Aquitaine on both side of Pyreness), but just enough to let its cultural and geopolitical mark on the region.
After that, I do think you'd end up with unyfing tendencies in Spain : it's not because you didn't have a concept of nation, that you didn't have the concept of common identity and what I would call a "common geopolitical horizon".
As for all peripherical regions, it eventually depends what happen elsewhere to say what happen in Spain.