[X] Accept Kukaku's offer of help.
-[X] Stay with her until it works.
Like Prospalz and others have said, Ichigo canonically
consistently excelled when thrown on the proverbial fire, and forced to do-or-die. He thrives in being trained into the ground. He should be able to make this 'until it's done' a short period of time, and in so doing both impress Kukaku, and learn something handy about himself and how to train his spiritual powers, that should be useful for trying to express train his schooling. Also, I'm almost always in favor of whichever iteration of votes advances goals the
fastest, since I'm a rather cynical fuck about quest lifespans, and the impact thereof on longterm plans.
And on the flipside, his motivation for all this definitely seems to suit this choice over any other; what even is the schooling
for, other than to get him a chance to do precisely what Kukaku's offering to do for him outright; contact his family and make sure they're not breaking at his absence? It's not
quite the same as reassuring them himself, but the fact she could even offer a message at all suggests she could afford some iteration of outright visit - or so I feel Ichigo'd reckon. Per this Kukaku's offer is more important than the Shinigami academy on the whole.
If you vanish for long periods of time, you have to explain it. Going off once a week? Not so much.
The point was that either way, Ichigo's on track to develop a
disgraced noble clan-style Shikai regardless, for obvious reasons. And further, even if he doesn't develop it out-of-house in a big notable breach of expectations/rules like staying away for weeks, he'd still be expected to
show his Shikai, whenever he develops it.
So unless you think he could find a way to disguise the way his Shikai looks on some fundamental level, that's no different between the choices.