There is very little guidance out there to address your question. So, in this case, I fall back on CPT / payer precedence.
CPT code 91038 is for
Esophageal function test, gastroesophageal reflux test with nasal catheter intraluminal impedance electrode(s) placement, recording, analysis and interpretation; prolonged (greater than 1 hour, up to 24 hours). Until all components of the code (placement, recording, analysis & interpretation) have been met, the code is not billable. So, I would code the date the physician review/interprets the test as the DOS.
This is consistent with Medicare guidance for a similar type of code, CPT code 93224 for
External electrocardiographic recording up to 48 hours by continuous rhythm recording and storage; includes recording, scanning analysis with report, review and interpretation by a physician or other qualified health care professional. This code is similar in that it is a 24 hour monitoring code and includes the same components (recording, scanning, review/interpretation). Medicare LCD
L34636 includes a billing and coding PDF (attached) which states that when billing CPT 93224:
"Use the date of physician review as the date of service (DOS)."
Hope this helps!