If I understand your requirements correctly, you want to run Tableau Bridge as a Fly Machine and have it connect to a PostgreSQL database inside of your org’s 6PN and do so using an IPv4 address.
If that’s correct, I’m thinking the most straightforward approach is to run either a PostgreSQL-specific connection pooler, or any vanilla L4 reverse proxy, inside of the Tableau Bridge’s Machine and have the Bridge talk to localhost:<whatever> (where whatever is anything but probably 5432 for consistency).
Doing so offloads the IPv6 resolution to your L4/pooler of choice and allows the Bridge to use localhost/127.0.0.1(or 0.0.0.0 if you want).
Concretely: use a process manager (supervisord, overmind, goreman) to run 2 processes: Tableau Bridge and any L4/pooler (PgBouncer, PgCat, Traefik w/ a TCP router) in a Fly Machine that can address your PostgreSQL database over 6PN (<app>.internal); have Bridge connect to your L4/pooler on localhost, and the L4/pooler proxy connect to your PostgreSQL database.
I haven’t tested this and YMMV, but it seems sound 