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Arrest information is a matter of public record, so website companies have a right to publish them. The issue arises when they post them and remove them for profit and/or when they are posting arrest photos of people whose charges were dismissed, or never prosecuted or who have been acquitted.
What I see is that the best way to get them removed is not to pay their fees, as sooner or later the photo will be picked up again when it's discovered by their bots and put back in their or in some other data base. The best way to combat something like this is to hire one of those reputation protecting companies to flood the web with positive things about you which would bury the mugshot photo by putting it towards the bottom of a google search, where nobody would be likely to notice it.
In terms of a legal suit, in general, this can be a gray area. If the record is accurate you're not being cast in a false light, and of course the photo would also have to have directly resulted in actual damage for you to prevail. However, several states have now taken up the issue and made it unlawful for a site to require a fee to remove a mugshot. Wisconsin does recognize a "right of publicity," which is a person's right to control his image and where and how it's being used if the people wh are using it can profit from it. That would give you a cause of action against mugshot.com to force them to take it down without paying a fee. You should consult with a local civil litigator who can assist you with this.