Mmmm.... I mean, it would, I guess, but I wouldn't recommend using both unless you can show a marked speed improvement over just using pgbouncer. A real connection is (probably) better than persistent connections, so you probably only need the one.

Now, I've not benchmarked this, so it's just a guess. Further, the specific performance characteristics will depend on your specific setup -- particularly upon where you have pgbouncer running -- so the only real answer is to benchmark and see.

New Relic does a good job showing time spent connecting to the db, fwiw, so you may want to give that a try.