If my opponent used Quaxly's "Apply Gel" attack against Tinkatuff on their turn, could I use Tinkatuff's "Pulverizing Press" attack and ignore the requirement to flip?
No, as long as the Apply Gel effect is on Tinkatuff you would have to flip heads in order to even start your attack.
Source: Paldea Evolved FAQ; TPCi Rules Team (2023-06-08)