S-76 EMS pilot reported after asking ATC to cross through their airspace on a medical mission; the controller asked the pilot to remain a mile north of the runway. Traversing the area; the EMS pilot needed to use evasive action to avoid an aircraft departing the runway.

Narrative

We were preparing to depart a hospital helipad en route to a scene picking up the patient at an airport. ZZZ1 is approximately 3 miles west of ZZZ and inside their Class Delta airspace and our route of flight would take us directly through this airspace. Prior to departing the helipad; I called ZZZ Tower using our medevac call sign and stated we were ready to lift from the helipad; we were medevac status en route to the airport. I stated our on-course heading and requested clearance through the Class Delta airspace. The Tower Controller said he had multiple aircraft in the pattern asked if I could depart and remain 1 mile north of the departure end of Runway XX. I was surprised I didn't get a clearance under medevac status to go directly through the Class Delta airspace on course but accepted the slight deviation to get moving. The problem was the small aircraft using Runway XX were still in their normal flow and one was departing and climbing out at the same time we were crossing the centerline on the departure end of Runway XX. To provide clearance from this aircraft I had to start an immediate descent. This provided clearance but we were still in close proximity to this airplane. Nothing was said by the Tower; and I didn't say anything either since the frequency was quite busy. I feel being in medevac status we were not given any priority handling and we should not be dodging aircraft in controlled airspace if ATC is looking out for us. On a side note; shortly after all this took place; we were canceled from this flight (which happens in EMS for multiple reasons) and continued south to our home base.