Have the organizations resolve the factual disagreement before relying on their letters. The application should explain the actual employer, assignment and professional services consistently, including a client placement where applicable.
Use a discrepancy sheet instead of choosing the nicer letter
Compare the named employer, worksite, reporting contact, duties and project dates in the documents. Highlight each difference and ask the person responsible for that fact to confirm the correct version. For example, a client may describe a six-month project while an offer mentions a longer employment relationship; those dates may refer to different things and need explanation. Keep the original records and the clarification together so the reviewer can understand what changed.
A payroll register shows which entity actually compensates the professional; a signed services agreement shows which one directs the work; a degree transcript or an entry in a licensing body's register shows whether the qualification attached to the listed profession is met on its face.