Air carrier pilot reported a Bleed Fail EICAS indication accompanied with a cabin odor during initial climb. Flight returned to departure airport.

Narrative

Shortly after takeoff; prior to finishing the after takeoff Emergency Checklist; we received a left bleed fail EICAS. We continued to clean up the aircraft and ATC issued a clearance to 17000 ft. Around 6;000-7;000 ft. we began running the bleed fail Emergency Checklist. While isolating the bleed for the Emergency Checklist; we immediately received a loud air noise under the floor with 'whooshing thuds' and light-moderate vibrations. Within 20 seconds we then noticed an exhaust or burned oil smell. The odor was not overwhelming but definitely noticeable. It was difficult to determine if the vibrations were aircraft or engine induced; but we never received an engine vibe EICAS. We quickly decided we needed to go back to ZZZ. While preparing for the air return; the odor and vibration seemed to oscillate from barely noticeable to noticeable until landing. ATC advised us they [requested priority] for us due to traffic priority. We didn't have the opportunity up to this point; as we were dealing with the malfunction. We notified Dispatch; Company and customers. We gave an ETA and code with the expectation of a normal landing and taxi to the gate. Cabin crew advised us they were experiencing the same vibrations and odor in the cabin. Upon landing roll out; the cabin temperature quickly became very warm. By the time we exited the runway we were both considering turning off the 'good' pack. But before we could take any action the right pack failed; showing an 'over temp' on the info page. We ran the Emergency Checklist for the right pack failure and taxied to the gate without cabin air and without further incident.