It's called SOLID color stain for a reason. It's a solid color, not semi-transparent, but it's still a stain. However, if these are permanent I'd use good cedar posts and use primer and oil based paints. Using a roller will work nicely.
When painting any pressure treated posts, plywood or even decking, you need to let the wood breath. Unless it's a good grade 2Xkiln dried, you'll have problems down the road if you use paint. Anyway, around here you do. The old saying if you wanted pressure treated painted... was to let the wood alone for a year and it will go through it's changes and just about dry out. Once it gets that greyish color, you can then paint it with any kind of paint.
The oil based paints will not allow the wood to breathe and this causes checking, warping and peeling of the paint as the pressure treated liquids try to dry. By using oil, they can't do anything, but using a stain... the various wet elements still inside can pass through and dry.
I would never use pressure treated for anything permanent. Unless you're lucky, they just don't look nice after a while. Cedar will look great and weather very well. I've had posts in the ground over 20 years without much problems. The biggest detriment to posts are people with weed-whackers. If cutting is needed right up against, use a small PVC sleeve at the base.