The rules support this, but check with your DM
Unfailing endurance says it gives as one of its effects “Endurance” which it then immediately describes as a “feat,” before going on to repeat the benefits of Endurance. While the rules lawyer in me would have liked them to flat out said “You gain the Endurance feat as a bonus feat,” or something, it beggars the mind to imagine anyone seriously attempting to deny that this is what the spell does.
And the Dungeon Master’s Guide rules for prestige classes state that requirements must be met in order to take the 1st level of the prestige class. They do not say anything about how you meet the requirements, aside from the fact that you have to have them all together before you get the level (that is, the benefits of the level itself cannot be used to meet them).1, 2 As I’ve explained elsewhere, they also do not say anything about losing the benefits of the prestige class if you stop meeting the requirements, and further because it specifically says you must meet them in order to take the 1st level of the prestige class, losing the requirement after you’ve taken that level has no effect.2 You can even continue taking levels in the class without ever meeting the requirements again.
As also mentioned in that answer, Complete Arcane and Complete Warrior have sidebars that seem to state the opposite as global truths about prestige classes. As described at length in that answer, this isn’t actually legitimate under the rules—a Dungeon Master’s Guide erratum would be necessary for that, and that one’s never been written—and the furthest those sidebars can be accepted is to apply that rule to the prestige classes in those books. Likewise, Book of Exalted Deeds has an explicit exception to the Dungeon Master’s Guide requirements rules for the prestige classes in that book. It’s possible that other books have similar statements, though off the top of my head I can’t think of any. Checking Magic of Faerûn, I don’t see any statement in it for its prestige classes or spelldancer specifically.3
Those are the official rules—but as Complete Arcane and Complete Warrior demonstrate, it doesn’t seem like the authors themselves were all on board with them. Many DMs aren’t either—they may not turn prestige class requirements into “pseudo-falling” mechanics where you can lose all your class features if you get ability drained or something, but they aren’t going to let you meet a requirement with a magic item or spell or something, either. And there can be gradations of acceptance—I myself am happy to see requirements met with magic items, but short-duration spells just really stretch my disbelief.4 But unfailing endurance lasts days/level, which is an incredibly long time. By the time you can cast it, it literally lasts a week. It is not at all implausible for you prepare and cast it once per week and ensure you have Endurance for the entirety of the time you would have spent training for your first spelldancer level.
So I would be good with it, personally. Check with your DM to make sure they are.
This is consistent with the Player’s Handbook rules on how to level-up, which says you have to pick your class first (i.e. before anything else that could allow you to meet any prerequisite you’re missing).
Both the timing of meeting the requirements and the effects of losing those requirements are different for feats (which also use a different term, “prerequisites”): with those, you can use the benefits of your level—choosing your feat is the last thing you do, per Player’s Handbook—but if you stop meeting the prerequisites, you cannot use the feat until you regain them.
One thing I haven’t bothered to check—but seems like it could be plausible—is a global Forgotten Realms rule in Forgotten Realms Campaign Setting or Player’s Guide to Faerûn. After all, those books add a bunch of deity requirements to a bunch of classes that don’t otherwise have them, so other tweaks to requirements are conceivable. Since Magic of Faerûn specifically references its prestige classes as being “in addition to those presented in the Dungeon Master’s Guide and the Forgotten Realms Campaign Setting,” any rules about prestige classes in general found in that book (and its 3.5e update, Player’s Guide to Faerûn) would apply to spelldancer. I don’t think either of them make prestige class requirements matter beyond taking the 1st level of the class, but it might be worth checking.
And then I remind myself that the “level-up ding!” is an arbitrary abstraction anyway, and there is no reason that someone should be expected to necessarily meet requirements at that exact moment—being able to meet them regularly and reliably ought to be quite sufficient, because really what you need narratively is to be able to train with the effect. But my knee-jerk reaction is that it’s unreasonable to claim to “time” your level-up in a narrow window of a few rounds or minutes.
