If I use Bramblin's "Ride the Wind" attack and flip heads, can I bring up Weavile and use its "Assaulting Hunt" Ability?
Yes, that'll work because the Assaulting Hunt Ability is still happening during your turn.
Source: Paldea Evolved FAQ; TPCi Rules Team (2023-06-08)

Relevant cards

Note: All applicable cards may not be listed below.
SV2_22
SV2_22
SV2_134
SV2_134