Belisarius gets the troops he needs and conquers Italia without the grueling and devestating gothic war of our world. After that justinian might go for hispanian or gaul - if he can conquer either, his successor would be in very good position to restore "mare nostrum". However, it's likely that the empire's treasury still be emptied as in OTL.
Assuming the gothic wars are avoided, Italy could quickly become a major source of tax income that could go a long ways towards keeping that treasury going. Look at North Africa IOTL- within a few decades you had an entire dynasty arising from it and it was always productive. If you get that out of Italy as well the Empire is in much better shape.
The next logical target is Spain, and considering the success Narses had on a shoestring budget an invasion on the level at which the Romans hit Italy with could be very effective. I think the best the Romans can realistically take and keep in the short and medium term is Spain on top of Italy and North Africa. Adding Gaul to that will simply stretch them too thin, even with more resources than OTL. And the Franks are much tougher and less divided foes than the Visigoths, even at this time.
Except if you can get enough disruption in "Iran" to almost completely stop land travel and have the silk road go to sea from India : it means Iran needs to fragment, be a warzone and a plague zone, so maybe get a deadly multi party civil war, add the plague on top and enjoy the peace that results for Rome

(that's what I've done in my TL for the late 2nd century AD)
I agree with
@John Farson- eventually you're gonna have the plague spread west. Things were too interconnected. But you could delay it through the means you're suggesting, and have it pop up at an "opportune" time for the Romans- and with better knowledge of what to prepare for.
A likely timeline could be something like this:
530s: Italy and North Africa are conquered and added to the Empire
540s: Integration of Italy and North Africa, skirmishes with Persia. Persia gets hit hard by the plague and wars with Rome pretty much end. Rome is (for now) safe.
550s: Invasion of Spain. Romans are fairly successful but do not conquer the entire peninsula, missing out on the mountainous north.
560s: Justinian dies, things settle down. Plague hits Rome, but is less brutal than OTL and serves to avoid civil wars. Persia is not yet recovered.
570s: Plague continues to devastate the known world, conquests are tested by Visigoths, Franks, Lombards, and other groups who the Romans manage to pay off or fight fairly effectively.
580s: Plague starts to die down. Major focus of Rome is now repopulating Italy and the Balkans and watching out for Persia, who is finally getting it together whether on the Sassanids or some other dynasty.
This probably butterflies away Islam and it's hard to tell when or if the status quo of Late Antiquity will be broken. I foresee a lot more conflict with Germanic groups than OTL, especially the Franks. The Empire is probably done expanding and will still run into issues as its treasury is strained and it has many threats to worry about. Groups to look out for also are the Berbers and Turks, who may fill in the void left by no Islam and do a bit of conquering on their own.