This isn't a simple fix and if you don't want to remove the bridge I would stop wasting your time on it. Once the top has warped and has taken on a new form, there's a good chance humidity alone isn't going to change it. I've never seen this actually fixed with just humidity. You need heat for it to take on a new shape and (hopefully) stay that way. Taking the bridge off lets you get enough surface and leverage to flatten things out. And, the bridge is sometimes part of the problem if it has changed in shape too.
I have had luck with the Thompson belly reducer. If you don't want to spend that amount at least look at the videos and see what they are doing. Sometimes, the wood will warp again if it's an issue with the board, bracing, and tension. But, if something in particular allowed it to warp, flattening it with heat might bring a permanent fit.
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