I travelled last year using my conditional green card. No hassles at all. I didn't need to present anything apart from my passport, GC and COVID test result.
Thousands-even tens of thousands- exit the Philippines every day and the vast majority of them don't have issues.
PH immigration officials target Filipino tourists who they suspect (rightly or wrongly) are going to work abroad illegally or who might be human trafficking victims. They don't bother hassling GC holders.