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The CRAs were incorrect as a matter of law and regulation in telling you to dispute directly with the party who made the inquiry.
The regulations implementing the direct dispute process clearly and explicitly exclude any dispute related to inquiries from the direct dispute process, period. See 16 CFR 660.4(b)(1)(ii). The reason, as explained in the rulemaking package for those direct disute rules, published in the Federal Register, is that inquiries are not information reported to a CRA by a furnisher, they are just the opposite. They are requests by a third party to view what has been reported by furnishers. Direct dispute relate only to dispute with a "furnisher," and not just anyone.
The rules also exempt other types of information from the direct dispute process that are or may not always be entered into your credit file other than by a normal furnisher, such as personal identifier information and public record information.
In distinction, disputes with a CRA are authorized under FCRA 611(a)(1)(A) for "any item of inforamtion contained in a consumer's file at a credit reporting agency," and is thus the only process for disputing whether an inquiry was made properly, which requires that the inquiree provided a statement of one or more of the stated permissible purposes under FCRA 604.
I would re-submit the dispute with the CRA, and cite both 16 CFR 660.4(b)(1)(ii) to show that they are incorrect in stating that you dispute directly with the party who made the inquiry, and FCRA 611(a)(1)(A) in support of the fact that inquiries are disputable via a CRA as being a covered item of information contained in your credit file.
If they still decline, you can submit a formal complaint to the CFPB and/or file a civil action under FCRA 616 for their willful noncompliance with the FCRA.