Early expansion of the Republic
The relative stability of the Republic in the middle of the 8th century was countered by the lack of decisive leadership that came with the turnover in leadership as the institutions of the new government began to coalesce. The old Republic could rely on the steady guidance of the Senate, composed of experienced leaders that had been in government for decades. When the new Republic was so young as to not afford that experience to its leadership, it could not avail itself of such guidance. Due to this, the early decades of the new government were dominated by a very timid foreign policy. That was not to say that the time was peaceful, as the territory governed from Constantinople turned into something of an armed camp.
As the 740s began to progress, the Roman state became increasingly assertive in its relations with its neighbors. The Umayyads, though facing severe internal strife, were far too powerful an opponent to seriously contend with. So, the Romans began to re-assert their influence over the slavic tribes that had migrated into the territory south of the Danube. The Bulgars ruled over many such tribes, but there were scores outside of Bulgar control, living in the highlands north of Greece. These were the targets of Roman campaigns, both military and diplomatic, of the decade. Overall, there were few setbacks, and the progress proceeded without any slowing. As the Roman presence in the territory grew, towns were re-settled with farmers from Anatolia and Syria, often by people who had been living under Umayyad governance until a Roman army showed up to rescue them.
However, the Bulgars, under their Khan, Telets, began to look at such expansionism with disfavor. They did not appreciate the growth of Roman power in their vicinity. Further, Telets had no blood ties to the Dulo clan that had ruled the Bulgars for decades, and which had maintained comparatively peaceful relations with the Romans. The Romans had had several treaties that had been signed when a Dulo Khan had been in power, such as the famous Tervel, who helped defend Constantinople in AD 717, and the re-settlement of the area was deemed to be in violation of those treaties. The dash-and-grab raiding that so characterized Roman-Bulgar relations intensified, and, soon, the Republic felt the need to push back firmly against these encroachments.
A campaign in AD 747 was led by one of the Hypatoi, Theophanes, to curtail these raids, and the army marched along the coastline of the Black Sea in order to bring battle to the Bulgars. However, still being an innately steppe nomad society, the Bulgars simply denied the Romans battle, and did little more than to harass them as they went. So, Theophanes began to attack the undefended towns and villages of slavs and former Romans, bringing their populations back to Roman territory to settle in regions in which they could more easily secured under Constantinople’s armies. Eventually, the Romans concluded the campaign, and marched back to the Republic’s territory. However, deciding to take the mountainous route back, Theophanes managed to lead his army into a trap.
While in a pass known as Varbitsa, the Romans were ambushed by the Bulgar forces, who had managed to prepare several earthwork fortifications in order to funnel Roman troops into a killing zone. The result was a slaughter, with almost half of the nearly 40,000 Romans in the army dying. Theophanes was able to organize a retreat, and the remains of the army made it south of the mountains, to safety. For his failure, Theophanes would not be returned in the next election. However, his advice was valued enough that he was a senior lieutenant to his successor, Christophoros.
This new commander sought to erase the humiliation of defeat by sacking the Bulgar capital at Pliska. This campaign took place in AD 748, and Christophoros marched straight through the same pass at which Theophanes had been ambushed, as a bold declaration of his confidence. This boldness was backed with some prudence, and the Romans maintained strict discipline throughout the march. When they reached Pliska, they defeated the small Bulgar army that attempted to defend the capital, though the bulk had, as usual, fled. So, the capital was burned to the ground, and the Romans marched back across the mountains, their point made.
This moral victory proved more fruitful in the immediate term than the previous campaign had, as it undermined the support that Telets had among the Bulgar elite. He was assassinated in the following year, and his successor maintained power for only two years, himself. What followed was an extended series of civil wars as the Bulgars struggled to maintain their order. Playing the various factions against each other, the Romans sought to maintain the disorder, and were able to secure the mountain passes that divided the two states from each other. Some minor campaigns were launched in the early 750s to take advantage, but the general policy was not to push the Bulgars against a wall and force them to re-unite under strong leadership. This policy enabled the Romans to conclude the consolidation efforts further south, and re-settle the region with productive farms. After that, their attention could turn further west.
The Lombards had enmeshed themselves into the fabric of Italy shortly after the reconquest of the peninsula under the Emperor Justinian. Sweeping aside the exhausted inhabitants, they became the new overlords of the region, just as the Ostrogoths had been before them. However, the Lombards rarely had the unity of rule that the Ostrogoths could claim to have through much of their rule over the peninsula. Broadly speaking, their territory was divided into three relatively independent realms. The north was ruled as the Kingdom of the Lombards, while the south was divided between the Duchies of Benevento and Spoleto. Scattered around and between these realms were the scraps of territory still ruled by the Romans, including Rome itself, Ravenna, Naples, and a still-minor town on the north of the Adriatic, Venice.
Though divided, the Lombards were still a major force, and had been steadily and methodically encroaching on Roman territory ever since their initial invasion. When one looks at a map of the period, the effect could be seen quite readily. On paper, Ravenna and Rome were connected by a strip of land that was under Roman control, but this was not always true on the actual ground, and the territory was never firmly in one camp or the other.
As the Republic began to consolidate and re-assert itself, the Lombards were one of the earlier targets of opportunity. After solidifying the border with the Bulgars on favorable terms, the Romans began to look towards retaking southern Italy, at a minimum. At that point, Benevento was hostile with the Lombard crown, and the Romans saw the opening they were looking for. Taranto fell back under Roman control in AD 756, and the Republic was consolidating its hold over southern Apulia, ready to push forward and secure the bulk of the duchy for themselves. However, the duke was able to make common cause with his Lombard brethren in the north, and the Romans were defeated outside of Bari. The Lombards counter-sieged Taranto and made a strong push to take the city back, but the Romans were able to hold out.
The Republic was able to secure peace under the current status quo, with only the southern tip of Apulia under their control. It was a victory, in the end, but hardly worth the effort expended to take it. The Roman Republic could be said to have won a Pyrrhic victory in its war for Taranto, just as Pyrrhus had against the Roman Republic, roughly a millennium prior. By this time, the Bulgars had re-united and were proving to be a more pressing concern than the Italian adventures. So, the Senate and the Hypatoi held off for the time, and began to look for more diplomatic options to enable their foreign policy goals. Fortunately for them, the son of Charles Martel, Carloman, sought assistance in asserting his power over the Franks.
End
So, in short: The Romans are able to conduct the equivalent of Constantine V's foreign policy, more or less on schedule, with marginally better results. They have a few more resources available on hand in Italy, largely due to the Iconoclast controversy being butterflied away. And, among the Franks, its the older son of Charles Martel, Carloman, rather than the younger, Pepin, that is rising to power.