Articuno, Motres and Zapdos!

Instead of using a probopass, Use a different Pokemon that traps them, such as crobat (Mean Look) or Dugtrio (Arena Trap) Probopass is too slow to hit before them. You'll also want to get their health as low as possible before putting them to sleep.
Once they are low on health and asleep, upon your next encounter, use a quick ball or ultra ball.

Here are a few options: 1: Get a Crobat and let it learn Mean Look and/or Hypnosis. Use the move then weaken the bird. 2: Get a Pokemon with False Swipe. Use that until the bird has only one HP left. 3: Use a Quick Ball or an Ultra Ball. The birds will still be hard to catch though, so buy a lot of balls.

Get a wobbafet I think that is how you spell it but it's ability stop the opponent from fleeing and running but if you manage to get it to attack it will probably kill the pokemon

I have great advice so use it.

Here's what you need:
A spare DS
An R4i with Another Diamond/Pearl/Platinum game
On the R4i the cheat that will give you 999x Master balls

And here's what you do:
Take the spare DS and insert the R4i into Slot 1. Put your Platinum/Diamond/Pearl game into your own DS. Play both games. On both games have a Pokemon ready to trade and go to the Union Room while your Pokemon that is ready to trade in the R4i is holding one of the 999x Master Balls. Trade Pokemon. Take the Master Ball off the R4i Pokemon. Repeat the trading part three times. Then run into one of the Legendary Birds and battle. Since the Master Ball never fails, use that at your first turn. Repeat the battle part for all Legendary Birds. Then you can catch all Legendarys if you know where they are!