Our inpatient group just had a discussion about this yesterday as our very large and very well-resourced outpatient practice has a hard time continuing LAIs that are started in the hospital setting. Even with patients that are generally higher SES with insurance coverage and an army of staff to assist with PAs and other bureaucratic hurdles, we were told that for most patients it's not possible to continue LAIs started in the hospital for one reason or another, and usually not for clinical reasons (e.g., lack of efficacy, side effects, etc.). I should note that these are second-generation LAIs, not first-generation, so if we stuck with haloperidol and fluphenazine perhaps we would have a different experience.
I mention that just to say that it is likely much harder to do this in the private practice setting. I do know that there are some docs in our community that routinely use LAIs - including second-generation agents - but they are relatively few and far between.
While things like storage and administration are worth thinking about, my understanding of things is that it is extremely difficult to even get to that point unless the patient has the cash to pay very high copays or the cost of the injection altogether. For what it's worth, when I was in residency we had some patients pick up the LAI from a pharmacy that we sent the prescription to and bring the whole dosing package to clinic where it was administered by one of our RNs if it was an agent that wasn't on the formulary. That seemed to work pretty well but introduces some uncertainly (e.g., actually picking the medication up, not losing it, not leaving it in a hot car for several days, etc.).