A fractional jet pilot reported they stopped their climb due to opposite direction traffic which passed 400 feet overhead.

Narrative

During this flight; I was the pilot monitoring in the left seat as the crew SIC (Second in Command). The PIC (Pilot in Command) was the pilot flying and was in the right seat. The seating arrangement was within company FAA approved policies since I have completed upgrade IOE for the left seat in this aircraft and was only awaiting a line check. While departing from Runway XX we were given heading 275 and maintain 3000 feet by tower. Tower also advised of traffic about 1 mile off the departure end of the runway which we had in sight. On departure; we climbed and leveled off at 3000 ft. behind and above the previous noted traffic. Just prior to 3000 feet we contacted departure and the departure controller gave us a climb to 8000 feet as soon as we leveled at 3000 ft; but did not issue a new heading. We began the climb and as I was repeating the climb instructions back to departure and I immediately noticed on our traffic display that we had an aircraft 1000 feet above us; at our 12 o'clock; less than 2 miles in front of us; traveling the opposite direction. I finished my radio transmission and looked outside and saw the airplane. As I finished my transmission a different voice on departure was giving a traffic alert; which I had recognized as most likely an ATC trainer that had stepped in to issue us the alert but I clipped the initial part of his transmission during my read back of our climb instruction. I knew we would merge with the traffic if we continued the climb so I pointed out the traffic to the PIC and told him to level off. As the PIC was leveling off; we got a TCAS RA also telling us to level off. I would estimate our separation from the other aircraft at about 400 feet. The ATC trainer asked if we received the heading of 250. I informed him we did not but we then turned to 250. We continued our climb to 8000 ft. after we were clear of the traffic and TCAS RA. The initial controller for departure did not assign us a new heading from 275; and gave us a climb to 8000 ft. which caused the RA and loss of separation. Since there were two distinctively different voices on the departure frequency; it was apparent that the initial controller was most likely in training; and the second voice we heard was the ATC trainer. It is apparent that the situation we found ourselves in; was due to an error made by the ATC controller in training.