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ChexSystems isn't the same as a CRA. They only have banking information. When a lender pulls your CRs they don't pull a report from them. It would just keep you from getting a bank account, even paid.
I would first contact BoA and find out why you were overdrafted in the first place and then not notified. If you find it is legitimate, you can pay it and ask BoA to have it removed. They may charge a fee to do this.
The same thing literally just happened to me--but instead of $375, it was $350 from 2011.
The Bank of America Recovery Department (their internal collections) still had the account, it wasn't sold to another CA so it hadn't reported on my CRs yet, I had only found out about it when I tried to open a checking account with a different bank and got denied.
Long story short, I called their department and asked them If I could settle with them... they said they could settle and close out the account and remove from ChexSystem if I settled with them for $200. I told them I only had $140 to pay toward this bill. The representative was really nice... she asked if I could do $150 and I just said yes... forward one month, it's been deleted off of ChexSystem, I got a letter from BOA saying my account was closed out, and all is well.
So you could probably get that $375 down to like $200 easily if you don't want to pay the full amount.
And BTW, you definitely want to take care of this while BOA still has it--because the representative on the phone told me they usually hold on to checking/savings accounts for 2-3 years, and then sell it to an external CA--if that happens, it will be reported on your CRs most likely through the third party. So act on it ASAP.
While Chexsystems deals with banking account information rather than consumer credit, they are still a consumer reporting agency under the FCRA.
You can file a dispute with them under FCRA 611(a) the same as any other CRA.