Depends on the chemical used. Today you should only run into ACQ treatment. The older CCA treatment is still used but only for certain applications and you shouldn't find it in a big box store. ACQ is copper based and that would be a problem for fish and generally not something filters remove. Your best choice would be ACQ Type D which is a common type.
The chemical doesn't leech out in great amounts, that would defeat the whole purpose of using it. Most leeching happens the first year. If you wash the lumber with slightly acidic water, like 4-5 pH, you can remove most of this first year leeching. Soap isn't a help, it's high pH.
There have been many studies on ACQ and leeching, but not a lot specifically on your case. This is not the kind of thing that kills all your fish after the first rain. More of a long term risk issue. Less of an issue if you're doing regular water changes, so depends on the pond too.
Here's something I don't think most people consider...the biggest danger will come from saw dust. Building the bridge near the pond, the saw dust gets into the pond. Or saw dust on the structure washes off and into the pond. The wood in the dust rots and all of the chemicals are left behind. Could be a much larger dose than normal leeching.
No absolute answer. More of a question of how much risk you want to accept. Someone with $20K worth of Koi in a pond wouldn't let PT wood any where near the pond.
Paint is no problem when dry.
If you're really concerned you could nail EPDM liner to the underside of the bridge so runoff went into the soil at the sides of the pond instead of in the pond. That would reduce chemicals by a great amount.
Big box stores may not carry redwood, cedar, etc., but I'll bet you have a wood specialty store that carries it...for a price.