A good inexpensive environment for a smaller goldfish is about 10 gal. Understand that a 1.5-gal-tank is basically equivalent to jail. It'll keep a small goldfish alive for perhaps about 6-12 months. I've used such a tank as a hospital tank.
That being said, goldfish are very adaptive and sometimes will surprise you with what they'll tolerate. In fact, in outside LARGE ponds, they can live below 50 deg temps. But what hurts their longevity indoors is crowding them in a tiny space. Goldfish as a rule will keep growing in size in larger tanks. I hope that the person who wouldn't sell you another goldfish is not the one that sold you that tiny 1.5 gal tank.
As I wrote before, find a new or used 10 gal and get a larger filter. Otherwise, a single goldfish in such a small tank is on borrowed time. That would be far more crushing for your young child.