I would review closely the below paragraph from the NCCI policy manual. Note they state if the criteria is met, that modifier 59 would be used with the pain block. The block would be reported under the provider who performed the injection.
4. Under certain circumstances an anesthesia practitioner may separately report an epidural or peripheral nerve block injection (bolus, intermittent bolus, or continuous infusion) for postoperative pain management when the surgeon requests assistance with postoperative pain management. An epidural injection (CPT code 623XX) for postoperative pain management may be reported separately with an anesthesia 0XXXX code only if the mode of intraoperative anesthesia is general anesthesia and the adequacy of the intraoperative anesthesia is not dependent on the epidural injection. A peripheral nerve block injection (CPT codes 64XXX)for postoperative pain management may be reported separately with an anesthesia 0XXXX code only if the mode of intraoperative anesthesia is general anesthesia, subarachnoid injection, or epidural injection, and the adequacy of the intraoperative anesthesia is not dependent on the peripheral nerve block injection. An epidural or peripheral nerve block injection (code numbers as identified above) administered preoperatively or intraoperatively is not separately reportable for postoperative pain management if the mode of anesthesia for the procedure is monitored anesthesia care (MAC), moderate conscious sedation, regional anesthesia by peripheral nerve block, or other type of anesthesia not identified above. If an epidural or peripheral nerve block injection (code numbers as identified above) for postoperative pain management is reported separately on the same date of service as an anesthesia 0XXXX code, modifier 59 may be appended to the epidural or peripheral nerve block injection code (code numbers as identified above) to indicate that it was administered for postoperative pain management. An epidural or peripheral nerve block injection code numbers as identified above) for postoperative pain management in patients receiving general anesthesia, spinal (subarachnoid injection) anesthesia, or regional anesthesia by epidural injection as described above may be administered preoperatively, intraoperatively, or postoperatively.