I've cleaned some pretty nasty brick patios with no chemicals at all.
On a few occasions I sprayed down some dilluted bleach (50/50)...........
There is one patio I clean every April for the last four years, this last time, I sprayed down some percarbonate deck cleaner and some detergent........and it cleaned up pretty good.
Did one last week that was really bad, but I used a surface cleaner and hot water on that one (not an option for you). That worked great..........but most of the time I don't bother with heat.
Keep in mind you dont have to get a brick patio to be totally flawless............anything you do is going to be a drastic improvement.
Just make sure you lower your pressure down a bit if you can..........you dont really need 3000psi to wash brick. Use a larger tip to cut the pressure down, or a dual lance wand.