Database connection pooling: PgBouncer vs application-level pools

we're running into a classic database connection problem with our current setup. we have about 50 application pods, each maintaining its own database connection pool of 20 connections, which quickly hits the 1000-connection limit on our rds instance. we've been looking at pgbouncer as a centralized connection pooler. the transaction mode sounds promising for minimizing connections, but i'm concerned about its limitations, especially with certain session-specific statements. we're also evaluating aws rds proxy. for those managing high numbers of database connections to rds or similar, what's been your experience with pgbouncer vs application-level pools vs rds proxy? what are the trade-offs in terms of performance, operational complexity, and feature compatibility?