I don’t have a definitive answer to the first part of your question, although I am very suspicious of the claim, since Ptolemy was a master at propaganda and rewriting history for his own purposes. While ruling Egypt, he wrote a history of Alexander’s campaigns that does not survive, but what we know about it from other ancient sources tells us that it was a highly propagandized account about how amazing Ptolemy was and how critical he was to Alexander's success. Especially after he seized control of Alexander’s body, Ptolemy made a major effort to link himself to Alexander, promoting the cult of Alexander as a god and even going so far as to mint coins with Alexander’s face on them. Creating a claim about him being the son of Phillip II would be well in line with Ptolemy’s strategy of linking himself tightly to Alexander.

I can give a better answer to the second part of the question, as to why he was not considered for the throne after Alexander’s death. Macedonian kings practiced polygamy; Phillip II had seven wives and Alexander already had three when he died. As a result, there was never a shortage of legitimate heirs in Macedonia. Illegitimate royals heirs were simply not part of the equation in Macedonian politics. If the Macedonian elites had been willing to accept an illegitimate child as an heir however, Ptolemy would not have been who they turned to first. About four years before his death, Alexander himself probably fathered an illegitimate child named Heracles with the daughter of the satrap of Phrygia (in modern day Turkey). Heracles’s name was brought up as a candidate by at least one of the generals in Babylon after Alexander’s death, but the idea was quickly discarded. The fact that Heracles was half Persian as well as illegitimate no doubt played a role in the decision to ignore him as a possible candidate for the throne. However, Alexander IV, the one legitimate child of Alexander the Great, was half-Bactrian, but was ultimately accepted as one of the joint monarchs alongside Phillip III. Heracles’s claim to the throne was only revived more than a decade later, after both Phillip III and Alexander IV were dead, and the normal rules of succession were clearly no longer in play.

It is possible that as a full blooded Macedonian, even an illegitimate one, Ptolemy might have been able to contest for the throne by playing that fact up against the half-Bactrian Alexander IV. The problem with this strategy would be that conservatives, especially the conservative minded Macedonian infantry, had already rallied around Phillip III as a full-blooded Macedonian royal. He could not compete with Phillip III’s impeccable blood claim as a mere illegitimate child of Phillip II. Another problem is the fact that the other generals would not have trusted him, believing him (possibly with reason) to be fabricating the claim in order to seize power. Ptolemy was not particularly well liked or trusted by the other generals in Babylon even when he was supporting the settlements (none of the generals trusted or liked each other very much). If he had made a brazen grab for power by claiming to be the rightful heir of Phillip, he would have unified all the other factions against him.

The other reason why Ptolemy was not considered for the throne is the fact that he did not put himself forward for the position. Ptolemy was present in Babylon when Alexander died, and was in the conferences that decided what was to be done about royal succession. In these conferences, he never put himself forward as a candidate for the throne, possibly because he did not believe he could make a claim to the throne stick, and possibly because he was not interested in the Macedonian throne. Nothing Ptolemy did during his life suggested he desired control of the entire empire. His actions in Babylon after Alexander’s death were very focused on stabilizing the empire and getting himself put in charge of Egypt. He initially backed neither Phillip III or Alexander IV as heir, suggesting a board of generals of officials be put in charge of the empire. However, he was not committed to this plan, and he helped broker the first of many peace deals a few months after Alexander died, ending the first round of strife with the establishment of the joint monarchy of Alexander IV and Phillip III. This put him in a position to demand, and be given, governorship of Egypt. Once he was in power in Egypt, he never showed much interest in the civil wars that were raging, except when they involved Egypt, or when they let him expand his control into the Levant. Even much later, when the death of most of the Argead royals left a vacuum, Ptolemy never made a claim to the imperial throne. In 309 BC, the last legitimate male heir to the throne, Alexander IV died. Ptolemy could have stepped in to to claim he was the next best heir as Phillip II’s bastard son, but he did not do so. He was always content to “just” be the King of Egypt. This was probably a wise decision, since most of Alexander’s other contemporaries got themselves killed trying to take control of the whole empire, but Ptolemy lived to a comfortable old age in Egypt.