What prevents the unobservable universe from being a continuously enlarging sphere due to the inflating bubble universes? Our observable universe being close to flat could support such a scenario?
Don't be mislead by the term "bubble universe". It doesn't mean the universe has spherical symmetry like a bubble. It just means the post inflation universe appears as a region within the larger bit that's still inflating. Any universe produced by inflation will be virtually flat as we discussed in Why does inflation (the inflaton field) push Omega down closer to zero (flatten the universe)?.
Having said that, you ask an interesting question and I'm not sure how the large scale geometry of a post inflation patch relates to to the geometry of the parent "multiverse". I don't think the post inflation patch "pinches off" from the multiverse, so I don't think it can have the topology of a sphere. That restricts it to the topology of a flat universe, though if it's big enough even a very small deviation from $\Omega$ = 1 will be significant.
As for the "multiverse", i.e. the bit that's still inflating, it could have any topology and we wouldn't know.