G550 Captain reported responding to an RA during a NMAC with climbing traffic while on a visual approach

Narrative

A Citation was climbing out and departing the ZZZ area Northbound and we were descending into the ZZZ area from the North after having been cleared for the visual approach to Runway XX. We slowed our descent when ATC advised the traffic but the Citation kept climbing until we were issued a 'LEVEL OFF' RA followed very closely by a 'CLIMB' RA. We followed the RA and passed about 400 feet over the Citation. Initially it seemed to me that the other aircraft did not follow any RA of their own or slow their climb to avoid us. We had been at a low power setting in the descent and I brought up the throttles to start the engines spooling in order to climb faster if the other aircraft did not slow their climb towards us and we needed to continue the climb to avoid the other traffic. The engines spooled up right as we passed the other aircraft and we started to level off. Because of this; we were very briefly above 250 knots indicated airspeed (around 260 knots). I brought the throttles back; we slowed to normal speed for the visual approach and landed uneventfully in ZZZ.Normally in the case of an RA; I leave the auto throttles engaged for speed control. In this case; I disengaged the auto throttles because it seemed that the other aircraft was not slowing their climb toward us or following any RA commands of their own and I thought we would need to have the engines spooled up to climb faster. The engines spooled up and we accelerated as I was leveling off from the RA maneuver. I think the extra power helped as we passed directly over the other traffic with a separation as little as 400 feet and I believe the other traffic was still climbing. I should have pulled the power back a little faster in order to avoid going above 250 knots. We noticed it immediately and slowed very quickly in order to restart our visual approach into ZZZ.