I feel you, mine also the same. I go to Parkway East, the paediatric urgent care there quite nice. The environment also not so intimidating, so kids won't freak out so badly.
U can try Parkway East. Both got very patient doctors who see small kids all the time. They used to my son’s drama already, still very calm one.
Parkway East Paediatric Urgent Care Centre https://share.google/I4AESeD71ByJgsS7j Baby & Child Clinic - Pasir Ris https://prbcc.sg/our-doctor/
Hahahaha same boat here. Maybe try to go earlier in the day when clinic less crowded.