Arcane spells and divine spells are not strictly defined anywhere in the rules. I mean, we know some things that definitely are arcane—in the core rules, those would be the spells cast by the assassin, bard, sorcerer, or wizard classes—and things that are definitely divine—for core, the spells cast by the blackguard, cleric, druid, paladin, or ranger classes—but we don’t know where the “boundaries” on them are.
One thing seems reasonably clear to me: only one of these should work. That is, it seems as though only one of these two options can be true:
Alternate Source Spell changes the type of spell for the purposes of Divine Metamagic. Thus an arcane spell prepared as divine can be affected by Divine Metamagic, while a divine spell prepared as arcane cannot.
Alternate Source Spell does not change the type of spell for the purposes of Divine Metamagic. Thus a divine spell can be affected by Divine Metamagic even if prepared as arcane, but an arcane spell never can be, even if prepared as divine.
I lean towards the former, but the latter is also plausible. I lean towards the former because Alternate Source Spell says “you can [...] prepare any of your divine spells as arcane spells or any of your arcane spells as divine spells,” and that to me means they have changed and so now whether or not Divine Metamagic applies also changes. Certainly, spells are prepared before they are cast, so by the time you go to cast the spell—and therefore, by the time you go to apply Divine Metamagic—the change has already been applied by Alternate Source Spell.
Now, all that said, I specified that only one of those options should work. I had a college professor who was fond of saying that “should” was the second-most dangerous thing you can hear an engineer say (after “oops”), and so it is here. I say that it should work like that because it makes logical sense to me, but... the rules don’t always make logical sense. We know for a fact that they don’t in any number of cases, even.
So here is the counterargument to what I thought made logical sense: Alternate Source Spell says you prepare a divine spell as arcane, and vice versa. But when you prepare a divine spell as arcane, nothing in the feat says it stops being divine. Nothing in the rules actually says arcane and divine are mutually exclusive. Just about everything in the rules implies it, but nothing says so explicitly—and even if something did, since Alternate Source Spell doesn’t, it could arguably be an exception anyway. Likewise, the word “Alternate” implies mutual exclusivity as well, but well, again, that’s only an implication (according to the rules, things’ names serve as identifiers and nothing more). And since we don’t have hard and fast rules about precisely what it means to be arcane or divine, there is room to argue for definitions of those boundaries that have overlaps.
Ultimately, it may not really matter: divine spells have all the advantages here. There aren’t powerful arcane-only feats analogous to Divine Metamagic. Only arcane spells risk failure due to armor. Divine spellcasters are restricted from casting spells opposing their alignment, but those are part of their class features, not an inherent property of a spell being divine—it wouldn’t apply to arcane spells swapped to divine with Alternate Source Spell. All else being equal, it is near-strictly1 superior for a spell to be divine rather than arcane (this is compensated for—over-compensated for, actually—by having the actual spells on the arcane spell lists generally be better than the spells on the divine spell lists). So if you have Alternate Source Spell, just make everything divine and enjoy all those benefits.
- I say “near-strictly” because it’s plausible that there exist some feats or class features that apply only to arcane spells, though for the life of me none are coming to mind at the moment. Certainly none as powerful as Divine Metamagic.