I started a timeline with this idea a while back, but let it die due to the lack of interest (
https://www.alternatehistory.com/forum/threads/spartas-illyrian-debt.140604/ )
I'd say that Sparta's main problem was that it was not geared to rule distant places, so its ambitions would remain rather local (even if it did campaign in Asia Minor) : I don't see it ever becoming a power larger than an unified Greece, which would already be a powerhouse ! But it had neither the ethos, drive or abilities to actually push much further than that, and would only be drawn into other conflits by Illyrians of Thracians raids, or by call for help from Magna Graecia, but without staying there long term because of Rome on one side getting them out of Italy (and probably Sicily) or because of lack of developpement of the balkan lands.
However, that same critique can be levied at the Roman Republic, but it still reformed.
Part of it I think comes from the Spartan idea that in some ways they weren't native or welcome (at least if I remember my reading correctly). So they had an element of insecurity about their own position.
Sparta probably COULD become an Empire, but it would depend on how they do it, and it almost certainly means addressing the Helot Problem.
A Spartan Empire and the Helots are either Fundamental, or Incompatible. It really can't be a mix here.
Either the Helots are part of the system, a permenant underclass that persists throughout the Lacedaemonic Empire, but are effectively brainwashed into obedience and service, or the Helot system needs to be replaced with a system of "Offset Levy" - or something of the like, where every polis and village is required to send either men to serve as Lacedaemonic soldiers, but they can pay goods in kind. This replaces the terror system with a more peaceful system of obligation, whilst still maintaining the Spartan Warrior Elite.
I think that could establish a Lacadaemonic Empire that was built around that system, you could effectively establish Lacedaemonic Barracks/Fortresses throughout, but otherwise have a tribute system be how soldiers are fed and paid. The Agoge system can stay. It was both prestigious, and a powerful cultural tool for transforming say a Massalian into a Lacedaemonic Warrior.
I think the system of Obligation is more stable in the long run, and would be a significant improvement for the Helots, even if the trust aspect needs to be dealt with. But a series of Lacedominc Warriors from the Helots (now no longer Helots, but (forgive the modern greek : ypochreoménos 'Obliged')) and the proof of good faith may well allow the momentum of the old systems authority to allow the new system to maintain its territory.
After that, you need a power dynamic. At the moment I'm just describing a tributary Empire. If it was more than that, we likely need a much larger conversation. A Defence For Tribute style Empire could work under Spartan ideals from what I can tell. Otherwise you need to somehow bring some sort of representation for either the Barrack-Fortresses, or other cities to the Spartan Constitution.