The war in Italy had nothing much to do with the conquests.
This whole idea that Justinian's western wars were hugely expensive and draining is one that urgently needs to be squashed, in my view. The reason they took so long was because Justinian didn't spend enough on getting the job done. When, after about 550, he actually put some serious logistical effort into the conquest, the Ostrogoths folded very rapidly.
Africa and Sicily were always net contributors to the imperial coffers, certainly until about 650 in the case of Africa (when the revolt of the Exarch Gregory seems to have done a lot of damage) and right the way through until the ninth century in the case of Sicily. Italy was less so, but then Italy had been in relative decline since the third century. Even on the Italian peninsula, though, the Romans kept the parts they needed well into the eighth century, and there's some evidence that Maurice was planning to evict the Lombards from Italy in the way he had done the Sclavenes from the Balkans. The disasters of the reigns of Phocas and Heraclius put paid to that, of course.
So, yes. No war in Italy does not directly lead to a stronger ERE. To do that, you need to butterfly away the plague of the 540s, and also keep the Iranians busy on their eastern front to stop the devastating war that raged with only a few small breaks for nearly a century after 530.
Anyway. Should Justinian not invade Italy, it'll be because Amalasuntha and her son are on the Ostrogothic throne, as secure Roman allies. At some stage, the Ostrogoths will convert to Chalcedonian Orthodoxy ("Catholicism", though that term shouldn't be used as most Christian denominations called themselves "Catholic" in the sixth century) probably in the second half of the sixth century. Once they do that, the chances of Roman attack become greatly lessened, although they'll likely still have to deal with the Lombards one way or another. Sclavenes and Avars may also have a pop at them, but the Ostrogoths were fairly tough.
The Ostrogothic kingdom will remain recognisably "sub-Roman" for centuries, I would think, moreso than the Franks did IOTL.