Hello and welcome,
The employer would typically be acting within their rights in seeking clarification of the need for the service animal from your physician.
However, if such additional information would not change the need for the accommodation, which appears to be the case, it does not appear to be a request made in good faith, but perhaps to harass you or delay the matter.
The employer must act in good faith in either granting your requested accommodation or seeking clarification, but there is not normally an unlimited right to seek more information with regard to your disability.
It would be best to retain a local attorney to reprsent you and communicate with the employer and file a complaint with the EEOC or state fair employment and housing commission if the matter is not promptly resolved.
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