I guess he would have spent more attention on institutionally rebuilding the Empire than Septimus Severus's "pay the soldiers and fuck the rest" attitude,
To be fair to Septimus Severus, that would be an awfully reductionist take on his political, military and institutional importance for the Roman Empire, the often quoted sentence being apocryphal at best, when his reign can be argued to have been the political apogee of the empire.
A lot of features we associate with the post-classical Roman Empire, although many could be understood as the result of a long evolution, can be either traced back either had cemented in his reign, namely : hereditary monarchy, final marginalization of senatorial elites to the benefit of the equestrian ones, stress on systematic judicialisation (remember that he had not a military but a legal formation), centrality of the imperial court over Roman networks, etc. Hardly a"fuck the rest" mentality.
In that, the importance of the military is clear (raising the wages for the first time since Augustus, giving them a better civilian status, using military structures to ensure civilian needs, etc.), but even more so because the traditional triple opposition between the Army, the Senate and the Emperor shifted to a dual relation in which the emperor was more and more identified afterwards as a military chief.
Another feature, which had clearly a negative and lasting influence over time would be the debasement of Roman coinage in order to face military and civilian expenses : although that wouldn't really be attributable solely to Septimus Severus as it was a steady course since Nero, his reign still saw an immediate drop in denarius actual value that himself or his successors didn't resolve, partly due to lack of any monetary theory at the time.
A different successor to the Antonine dynasty might have not underwent such practices (universally loathed in Antiquity because considered akin to counterfeiting) or at least not to the same extent. I do wonder if they'd have done a better job as Septimus did in re-establishing stability if so : the worst case scenario for the empire at this point would have been a longer, drawn-out, civil war.
Regardless of our perception of the individual, there's little way to refuse him being one of the most influential Roman emperors who brought as much stability could be hoped for in the eve of the Third Century Crisis. Blaming him for it would be comparable to blaming Richelieu or Louis XIV for the French Revolution. (And even that comparison fells short, as, overall, the reigns of some of his successors as Caracalla was actually stable enough).
what happens if the Year of Five Emperors is averted and Pertinax is able to establish himself post-Commodus.
The core of the problem resides in who was Petrinax and how he was proclaimed emperor.
Before Commodus, he was a distinguished civil servent in the city of Rome, but his military record was far from stellar as he barely avoided a mutiny and an assassination in Britain during his only real military command.
Having a priori no hand in the conspiration that killed the emperor, he was designated as his successor in order to legitimize the murder thanks to his good reputation and senatorial standing.
That...doesn't spell that a good of a political support, and rather vulnerability in face of the army and provincial networks : you mentioned the possibility of bringing in more loyal troops, but while loyalty to the imperial "legality" or the empire was real, loyalty to a person had to be earned trough a direct relation with troops (current or past), something Pertinax most definitely did not had.
The reality of it is that Pertinax was in a very vulnerable position at this pioint, not able to refuse the throne, not able to compromise with the praetorians, not able to go against the Roman supports...
It was all very well for Septimus Severus to come in as Pertinax' avenger without actually having to deal with the same issues.
While Severus was definitely not bound to succeed, Pescennius Niger had a fairly strong powerbase himself and so did Claudius Albinus, but Pertinax (or Didius Julianus, for that matter)... I must admit I don't really see *how*, at least bound realistically in the confines of the year 193. (I do think it could make an interesting AH, still).