Maybe I am confused, but the IVR is analog and the pharmacy stations are on the Mitel... Where do the SIP trunks come into play that you want to flash on them?
The call would come in as SIP, then be converted to analog POTS buy the system and delivered to the IVR... If the IVR doesn't handle the call, it will do a hookswitch flash on the analog extension, dial an extension (or Hunt Group) and disconnect. The trunk type in this case is not relevant, unless I am missing something.