I did it. It took me three years, but I did it.
That is, I stopped playing since I couldn't beat the Snagem boss. I restarted twice just to get to that point and fail again, but the third time I made it all the way through, and eventually got Ho-oh. Of course, Suicune, and to a lesser extent Raikou and Entei, was key.
In double battles, try sending out Raikou and Suicune. Raikou, hopefully the faster of the two, uses Rain Dance and then Suicune uses Surf. If that doesn't kill all of them, Thunder's now 100% accurate.