Seems like a pretty big assumption re: Livia's ostensible powerlessness. Aside from the Senatorial historians like Tacitus who are our sources for the claim that she went around assassinating Augustus' alternative heirs (or even Augustus himself), she was pretty widely respected as Augustus' wife, the ideal Roman matron and a woman of some independent means (Augustus allowed her to run her own finances & businesses) in her own right. (If we are to believe those historians' rumors, then she certainly would not have even waited for Augustus to die to plot Arminius' demise) Certainly she must have maintained positive relations with at least some Senators, as they tried to give her the dignity of Mater Patriae (stopped only by the opposition of Tiberius himself) later in life. Her relationship with Tiberius doesn't seem to have collapsed until 22 AD; and I think she might have more success in persuading Tiberius to take up the Principate if the alternative is literally, as far as they & the rest of the Roman elite can tell, some random Teuton. I think she's absolutely a viable player in a situation this bizarre, especially with Agrippa Postumus already pretty much done for.
Varus has distant marital ties to Augustus - his first wife was Augustus' niece and his second was Augustus' grandniece. He would sooner make a play for the throne himself than support Arminius, even if they are friends, especially as I don't see anyone else stepping up to join him if he commits to such a course. In fact, friend or not, I don't see him trying to make Arminius emperor unless he already thought the latter was a god.
Regardless, once again I must reiterate: far mightier barbarian warlords commanding the majority-barbarian Roman army in the Western Empire's twilight years could not make a play for the imperial diadem. Not Arbogast, who straight up staged the suicide of the legitimate emperor at the time (Valentinian II) but then tried to make a Senatorial ally of his emperor rather than go for the purple himself, nor Ricimer, who made & unmade multiple emperors in a row.
Even Stilicho, a half-Roman general with a decades-long proven record of diligent duty to Rome (unlike Arminius who's joined the Roman army in 1 AD at the earliest, and thus would have had at most under a decade & a half of such service), who repeatedly saved Rome from Gothic invaders (whereas Arminius didn't even have any wars on the scale of Alaric and Radagaisus' invasions of Italy to fight in) and who was directly married to a Roman princess himself, couldn't pull it off. The mere possibility of him putting his son (who was not only 3/4 Roman but also as closely related to the last uncontested emperor, Theodosius the Great, as Varus' second wife was to Augustus) in the imperial line of succession gave the Italo-Roman elite the pretext to kill him & his entire family in a moment of vulnerability.
Not even Odoacer dared try it, and he was the one who extinguished Rome's light in the West altogether twice - first by disposing of Orestes & his puppet usurper Romulus Augustulus, then by crushing Dalmatia where Julius Nepos had held out. Instead of making himself emperor, he gave the imperial regalia to Zeno and submitted to the authority of the only Romans left (for all the good that would do him a few years later).
To be blunt, I don't see any fucking way whatsoever that Arminius - a barbarian who wasn't born on Roman soil, is barely halfway through his 25-year term of service in the Roman army, and has no real ties to Augustus or his inner circle beyond that friendship with Varus - would have been allowed to become Emperor in 14 AD, when Rome was still the unquestioned #1 power in Europe and very far away from collapse. Not with other claimants who'd be vastly more legitimate in Roman eyes running around (Arminius' sole claim to being Augustus' heir in this scenario seems to have been that Augustus said so, and plenty of other emperors' last wills have been disregarded despite making vastly more sense) and a Senate that both isn't totally toothless yet & still has some memory of a pre-Principate republic. Arminius could just declare himself Rex Romae and it still wouldn't have worsened his position, because after all he can't sink his chances below rock bottom.
You don't necessarily have to have Tiberius still succeed Augustus, but Arminius? I'd put his odds at where Jesus' would be if Tiberius had named him heir and then died in 30 AD.