Captain reported concerns over operation of an aircraft with an out of date NAV database that should have been deferred. The aircraft was later given the correct MEL as required.

Narrative

On Date; I received a phone call prior to shift change from the day pilot; telling me he would be slightly late to meet me because he was in ZZZ1 on a patient transport. He also advised me; as a heads up for my oncoming shift; about an issue regarding the Garmin 725 GPS in our airplane; Aircraft X; concerning a NAV data change/update switching to Jeppesen from the Garmin NAV data in that was previously installed. He told me that new NAV data card wasn't communicating properly with the terrain and obstacle information on the other installed Garmin card making the TAWS inoperable. He said he researched and discussed with our Manager; and that the issue could not be MELed to their understanding but that it was legal to fly as long as the NAV data was current and functioning properly. I expressed that their findings were not consistent with my interpretation of the regulations; MEL; or past experience with how the same scenario was handled with a previous part 135 employer. I suggested that he leave the airplane there and fly the spare; Aircraft Y; back to ZZZ and let Maintenance resolve it. He assured me that they deduced it was legal to fly and that the preference from Maintenance was for Aircraft X to go back to ZZZ and for the mechanic based there to fix the problem the following day. I deferred to their decision as Name is both a seasoned pilot and Aviation Maintenance Technician; had already flown a flight that way; and Name 1 is our Manager; though in truth; this did not sit well with me. Towards the end of my shift the following morning; Date 1; I received a flight request to transfer a patient from ZZZ to ZZZ1. We departed ZZZ at XA:25 with the patient and med crew onboard on an IFR departure/flight plan to ZZZ1 in VFR conditions; and returned to ZZZ with the med crew onboard at XD:02 also on an IFR departure/flight plan in VFR conditions. Although both flights were IFR and thus terrain and obstacle protected by ATC; the TAWS system was indeed inoperable for the duration of both flights and; as would become clear later; not legal without an MEL sign off. Name was there to meet me for the pilot shift change briefing and his preflight. I communicated to him that it was my belief that we needed to have Maintenance resolve the TAWS issue before he and I accepted any further flights and he agreed. The ZZZ assigned company mechanic could not resolve the problem; but he and Name did find from escalating the issue to various company maintenance superiors that it was indeed not legal to fly unless an MEL sign off was processed; yet was actually eligible for a 2-day MEL; which is what the mechanic then completed.I don't think Maintenance should have even released the airplane back in service to the pilots after switching out the NAV data and having this issue; knowing full well that the TAWS system was inoperable. That said; as the pilot in command (PIC); I should have asserted my belief and interpretation that the airplane was not airworthy in its current state and escalated the issue myself to get a resolution; instead of resigning myself to the conclusion that the previous pilot and Manager came up with.