It would be fine if you arrange the orbits properly.
This system wouldn't necessarily be problematic if you can maintain stability via particular orbital resonances. We've discovered numerous compact systems whose stability is believed to be maintained on long timescales in this manner. Kepler-90 is a great example of a compact system kept in place by resonances; it's a roughly 2 billion-year-old Sun-like star that harbors eight planets within $\lesssim1$ AU. The outermost (and most massive) is roughly 2/3 the mass of Jupiter, orbiting right around 1 AU.
The system you're proposing is quite similar - presumably just with a gas giant in the middle, not in the edge, and with an increased spacing of planets. You might be concerned by this rearrangement, but note that the gas giant is only $\sim0.3$ AU away from the next planet over. Some models predict that the Sun's habitable zone is $\sim0.5$ AU across, making it quite feasible to have a similar spacing here (though of course spacings much less than $0.3$ AU are still rather stable).
L.Dutch cites the asteroid belt as a reason for worrying about stability; I'm not as concerned. The asteroid belt was formed in part due to unstable resonances relative to Jupiter's orbit, rather than stable resonances. The Kirkwood gaps are one set of notable results. But bear in mind that the presence of an additional gas giant can also lead to stable resonances, as I noted above, and may actually make it easier for compact, stable systems of terrestrial planets to form (see Hands & Alexander 2016).