I couldn't put this part in due to space limit in questions but I'm gonna use the FASTEST Pokemon that can meets the requirements(also that Pokemon being faster than Raikou would as I got Entei paralysed.) Also if no Pokemon fits then at least a Pokemon to stop Raikou from ending the battle so that I can paralysis it.
Hopefully this will meet your requirements:
I would personally suggest Wobbuffet for your pursuit against the legendary dogs. Wobbuffet's Shadow Tag Ability would prevent Entei/Raikou from running away again. In HG/SS, Wobbuffet can be found in the Cerulean Cave in Kanto or the Dark Cave in Johto. Although battling with Wobbuffet would be somewhat difficult, I'm sure you could pull it off. :D
Otherwise, fast-moving Pokemon that can learn Mean Look potentially includes: Crobat, Gengar, Jynx, etc. I'm sure you can find something that suits you.
Good luck, and may the odds be ever in your favor ;P