Likely requires IGT tracking and several difficult manips (both making them and executing them). These manips have not been made, and there is still some question as to their viability in RTA runs. Uses Mega Punch route, which is explained here: https://pastebin.com/3YwK4qPG

BUG CATCHER
3-second swap
97-second fight
16 seconds of random encounters before second Bug Catcher
avg. 14-15 damage taken

PIDGEOTTO
23-second manip
11-second encounter
??-second fight (<77 seconds saves time)
5-second loss from lower level on Metapod
3-second loss from Mega Punch route
avg. ?? damage taken

CONCLUSION: Probably not worth it if the battle cannot be manipped at all. However, manipping turn 1 paralysis could save a LOT of time (25+ seconds). Ideally we'd find a cluster of frames with favorable outcomes (e.g., frame 1 Sand-Attack fail, frame 2 Gust into TS paralysis, frame 3 TS critical).

A perfect fight would be:
Turn 1: Gust, TS paralysis
Turn 2: TS crit, Gust (Pika dies)
Turn 3: Tackle (Pidgeotto dies)

That fight saves about 45 seconds over fighting the Bug Catcher.