In OTL, Octavian ordered his soldiers to kill Caesarion and Augustus' successor would be none other than his "stoic and unpopular" stepson Tiberius. What if Augustus had decided to spare his cousin and named him as successor, viewing him as the only and rightful heir to Rome, instead of his stepson? Best result: No Caligula.
As far as Octavian is concerned this is totally ASB. There might be ways to bring Caesarion to the Principate, but not with just this POD. Octavians justification for going to war with Anthony, is that what he was doing was so un-Roman, and was totally unacceptable to any right thinking Roman, and more ever under Roman Law, he, Octavian was the rightful heir. Hes not going to undermine his whole claim to power.
Furthermore, as Paterfamilias and Pater Patriae, Octavian needs to promote traditional Roman virtue to conciliate reluctant conservatives, as he not so subtly strips power away from them.
Hes not going to let the bastard son of some poxy doxy, decadent eastern despot, stand at the head of the Roman Republic. Thats just not the way Rome worked back then. Not unless he wants to find himself lying in the Theartre of Pompey, with a knife in his back