Hmm ... I think, it is possible we could go with the interpretation that Precocious Apprentice negates itself by its own qualifiers.
Obviously not intended, but it's an interesting thought exercise, while we're working to come up with a consistent interpretation of what was intended. Since ... you have a spell slot, and can cast at least one 2nd-level spell spell, and I'm not convinced that there is anything separating it from casting other than fiat (even official fiat).
(Haven't been following Solo's secondary argument too well, but ... In theory you could cast all your 2nd level spells with caster level 2 with one optional ruleset at least - the trait, Spellgifted, could decrease the caster level of all of a wizard's spells by 1 if they selected carefully against their own build choices. Unless you actively can't take 3rd level spells of those schools until wizard 6. This might be true. Now I am rather curious.)
At that point, you lose the benefit, but still retain the bonus 2nd-level spell slot, and likely still must succeed the level check to cast the particular spell you chose in the first place, if you know it, since your level is still not high enough to cast it.
You might not have any 2nd level spells known. I think there are some methods that give you spells known beyond what your level normally would, which may have been referenced in this thread already, but I am not familiar with them; failing that, your second level spell slots would be usable only for such mechanisms as powering feats or possibly spellthieves.
... Interesting that the feat does not obligate you to keep the spell. Interesting also that it does not give you the spell known at all. How is a sorcerer even using this feat, as written?
Interesting that apparently all you're actually losing upon being able to cast 2nd-level spells appears to be the penalties.