Yeah, they still called it the Republic, because the first Emperor was smart enough to want to keep its changes hidden. Indeed, most of them seem to've been quite clear that there had been a change, and even wistful about it, because they knew their Republic had had a better run at conquering people and prospering than any monarchy in history. But that doesn't mean our habit of using different names isn't right. If America added a King position with complete authority, would you call the result a democratic republic?
The Republic was still called the Republic because it still was. Post-Octavian didn't really end the Republic, he simply reformed it. Octavian was able to end the rounds of civil wars started by Marius and Sulla by reforming the Roman constitution. He monopolized military power in his hands and with this monopolization bent the whole system of Roman office-holders to his will.
It lasted because it was necessary. The rounds of civil wars showed the old Republican system had outlived its usefulness, and was no longer sufficient to deal with the large territory that Rome had to administer. A centralized administration with a stream-lined decision making process was the answer, and Octavian crafted what that decision making process would look like.
I don't see why Rome would agree to the kind of radical reforms that you are proposing. Octavian was only able to impose his own reforms after almost two full generations of civil war, that is only able to impose his reforms after the leadership of both factions had been killed off and the Roman political class convinced that one man rule was the only answer.
The Christians'd be handled by the religious tolerance law, which is really a much more characteristically Roman Republican way of handling religion. So you'd get, in the long run, a healthy Christian religion, but no more danger of theocracy or intolerant Christianity empowered by elected consuls than the US faces.
I think that Christianity arose from the terrible condition that the Roman Empire found itself in during the 3rd century crisis. Plagues killed huge numbers of the population, enemies ravaged the empire both within and without, as Roman Legions and their would-be Emperors fought each other and barbarian tribes menaced the borders. Basically, Octavian's monopolization of military power as the key check on chaos had worn off, and a new control had to be found. Christianity offered an Empire-wide organization who could impose uniformity on an Empire that must have seemed on the edge of collapse. If the Emperor supported Christianity, he could have a thousand bishops all telling their congregations: "Be loyal to the Emperor, the Emperor is a friend to the Church." That was a powerful selling point. In a time when the legitimacy of the Empire was being put to the test, the Church offered its voice to keep the Empire going, in exchange for the Empire's protection and promotion.
Now in your ATL maybe the exact same political conditions don't exist, but the plagues and what not will still shake confidence in the system. Huge numbers of deaths, threatening tribes, those are conditions for a tyrant to emerge.
I didn't talk about the continuing Republic vs its neighbors atall, alas. I was hoping to get into conquering stuff a bit more, but it's clearly not happening. Generally speaking, the Republic did much better at adapting quickly than the Empire. Polybius pointed out that they turned from naval novices to dominating the Med pretty quickly, and didn't just adopt Greek military ideas, but improved on them enough to conquer the Greeks.
By contrast, Imperial innovation moved slowly, because innovators had a way of seeing their careers killed unless they were Emperor, in which case they became too busy to innovate. History shows the Empire failed to adapt to their meighbors' tactics, so as soon as the last set of Republican tactics went obsolete, they went from growing to shrinking longterm with the occasional regrowth until cannon finally did them in. I got as a Christmas gift when I was a teenager a set of European maps of many differing dates, and it shows those trends clearly.
One difference I was seeing was that instead of the second civil war, Caesar'd likely conquer the Germans. The OTL Republic and then Empire IMHO had their problems conquering the Germans because the turf was bad for Roman reprisals - crossing the mountainous Alps, which makes it at least twice as hard as it had to be. I'll bet Augustus' failed expedition was also lazy about finding allies and scouting and mapping. Caesar, who started his successful OTL incursion for ,German R-E-S-P-E-C-T, was smart enough to start by bridging the Rhine instead of crossing the Alps, and smart enough to find allies and scout.
Yes, lots of them certainly would get sick and die of plagues, in the same kinds of proportions as OTL. But it'd be hitting a stronger body politic, which would be much smarter about adapting. They'd probably encourage more immigration in a smarter way than bringing whole groups of Goths in and let them live and fight Gothic-style, under Gothic leaders. They'd make smarter alliances with their neighbors than the Empire. Most importantly, because the Republic'd have military innovations to match the nomads and more, unlike OTL, it'd face fewer bad military consequences. It'd probably come out to being more like having to stop expanding for awhile.
The Empire didn't get any better because there was not a whole lot better that they could do. The Legions had the best training and technology that Rome could get its hands on, in terms of both military and civil technology.
Allowing the Germanics to cross the frontier as complete tribes? Really? Allowing barbarians into your lands fully organized under their own leaders was what happened OTL to the West. The result was the fall of Hispania to the Visigoths, North Africa to the Vandals, and Italy to first Odacer and then Theodoric. In other words, the dissection of the Roman West.
In terms of territory, the Republic's territorial expansion was driven by the same forces that created Octavian's reforms. Rounds of territorial expansion provided the capital for the conqueror to impose his political will on Rome. Another faction would oppose him, and presto, civil war. Marius-Sulla was funded by foreign conquest, as was Caesar-Pompey and even Octavian-Antony could be chalked up to Antony's access to Egyptian wealth. Finding some way to keep the conqueror and the conquered wealth separate would seem a good idea, but it wouldn't be popular in a system where wealth is necessary to wage political campaigns, and the only real sources of new wealth is foreign conquest.
Basically, you need to give the Republic's political leaders some other way to raise money besides taking it from the conquered. In addition to finding some kind of compromise on the class thing. And none of these reforms will happen without a foreign military defeat or as result of exhaustion from civil wars. Having the seen the results OTL of what system exhaustion from civil wars brought about, why not try military defeat?