Translate the work into consulting-relevant behaviors rather than focusing on the internal label.

Highlight ambiguous problems you structured, executives you influenced, functions you aligned, analyses that changed decisions, and implementation you drove after the recommendation.

Where possible, show that you worked across business units or with stakeholders who had different incentives, because that approximates the influence challenge of client work.

Be transparent that external client service is new. Then explain why your track record of solving unfamiliar business problems and influencing senior stakeholders gives you a strong base to make that transition. For a related reference, see experienced-hire recruiting.

The advice I got was to avoid defending the first idea just because I said it first. Updating your view is part of the case.