If you do sue, you sue both companies. If the policy was sold on Geico's website, then American Bankers is either a subsidiary of Geico or it contracts with Geico to sell products on Geico's site, using Geico's reputation. I can see the advantage of the relationship with all the advertising that Geico does. Trial lawyer's rule of thumb: sue all possible parties and let the defendants sort it out.
I hope you are not intending to do this yourself . An insurance bad faith case is not a case to be tackled, pro se.
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