Perdiccas was from the house of Orestis, which was a high ranking princely line, but thats not the royal line of the Argeads. Leonnatus the bodyguard actually had the closest blood relation, through Alexander's grandmother, while Antipater was a much more distant relative (to a similar degree of Perdiccas likely) and Neoptolemus (Alexander's armor bearer, mostly famous for his rivalry with Eumenes) was related to Alexander's mother (possibly her nephew). And Alexander had nephews in Epirus through the line of Cleopatra and his uncle (also named Alexander). This doesnt help any of them though, since a distant relationship isn't enough to actually claim membership the to Argead line.
I knew Perdiccas was of the leading house of Orestis, and not the Argead line, but I thought I remembered reading that he was distantly related to Alexander. That seems to have been a false memory though, since I checked with a couple of my books today and found nothing. Derp.

Must have been confusing him with Leonnatus; a reminder to check sources before claiming something I'm not certain about.
Now that said, I do still think that Perdiccas being crowned is a legitimate possibility with a complete lack of other legitimate candidates.
I think you are somewhat underestimating Peddiccas' popularity, since although he was opposed by Meleager and the infantry, he also rose higher than any other man in Alexander's army post-Hephaestion, over a number of other very popular, very charismatic personalities. That's not to say he was loved by the infantry (or even the cavalry), he was no Craterus, but he was at least very well respected as a competent officer and someone who could be calm under pressure. Claiming that he was unpopular and unliked at this point seems like hindsight bias - I can't recall anything I've read that suggested that he had this reputation to the army in general before Alexander's death. Of course if you can counter this, please do.
The split between the infantry and cavalry came because the infantry chose Arrhidaeus and the cavalry (in the guise of the bodyguards, all cavalry officers, plus Meleager and Nearchus) chose to see if Roxane would have a boy. Which mostly happened because of racism - Alexander's child with Roxane would be half-Asian; the conservative infantry wanted a full-blooded Macedonian to rule them. Meleager took advantage of this rift to try and challenge Perdiccas, which of course went poorly for him. Meleager tried to execute Perdiccas, not the infantry, and Perdiccas took advantage of the rank and file's loyalty to him and the other chief officers to escape execution. As I understand it, the mutiny against Perdiccas wasn't a rejection of Perdiccas so much as it was a rejection of Alexander's attempt to unify the Macedonian and Persian ruling classes (with the Persians of course as the junior partners).
Now this brings us back to Herakles. Assuming people take him seriously in this timeline as a candidate, Herakles' mother Barsine was half-Greek, half-Persian, meaning Herakles would be a quarter-Persian, and although he does have the advantage of being out of his mother's uterus at Alexander's death, he has a lot of the same disadvantages as Alexander IV had, on top of being illegitimate. I can definitely see the infantry rejecting him even if the little council decides he's the best choice. They would want, if possible, a Macedonian to rule them.
And, in this scenario with the Argead line dying out with Alexander, and if we disqualify Herakles, who else is a better candidate for the throne than Perdiccas? He has the signet ring, he is the highest ranking officer in the army, he's a respected military leader, and he has royal blood - not Argead, but of Orestis. He has the most troops loyal to him out of any of the officers. Curtius claims that IOTL there was a proposal that Perdiccas be made king - which provides some small sort of legitimacy for this idea, even if we don't know whether it actually happened or is simply a Roman fantasy. The officers might decide that crowning him is the best way to maintain stability, and the infantry I think would be more likely to go along with him than with Herakles. This isn't to say that it wouldn't fall apart at some point, because I think it would, but I think Perdiccas being made king is a legitimate possibility, even though he's not distantly related to Alexander as I mistakenly thought.
Now, having thought about it a little more, maybe the most likely possibility is to have Antipater and Craterus rule in Europe and Perdiccas and Leonnatus rule in Asia like the initial theory for the regency went IOTL, except without a king to be ruling in place of - unless they decide they're acting as regents for Alexander's ghost, which is actually not impossible now that it crosses my mind.
Anyways, there are a lot of possibilities, but I think all end in roughly the same result - a bloody, messy civil war, not unlike our timeline. It's really hard to avoid unless you have grown up, legitimate children of Alexander succeeding him. Things will fall apart, whether Perdiccas is king, or there's a regency for Herakles, or the "board of four".