After around ~240 throws, I finally caught Entei in a Poké Ball. I used Ariados with Quick Claw, which had a 20% of not letting Entei flee, followed by Spider Web that prevented it from fleeing directly, followed by BPing to Electrode which also prevented it from fleeing via Roar. I was able to use about 60 Poké Balls a time. So it took me around 20 encounters to finally succeed.

I’ve counted 172 Pokémon / 421 Pokémon in my Poké Ball Genderful Living Pokédex after some more breeding/evolving following that. So it’s now just a case of breeding/evolving wherever possible, including doing the in-game trades before I start my journey on FireRed. Maybe I can do a level 5 living dex on there.