PA-46 Malibu Meridian pilot reported fuel transfer pump malfunctioning during departure climb resulted in altitude overshoot and airborne conflict.

Narrative

The aircraft is a low wing single turbo prop which requires the fuel be managed within the 10 gallon maximum imbalance limit which requires constant attention especially while climbing as fuel consumption is about 45 gph at a climb rate of 1500 fpm. While climbing through 13;000 I noticed that the low fuel annunciator for the header tank (the wings have separate electric pumps that transfer fuel from the selected wing into a header tank that feeds the engine)was illuminated as was the one for the wing transfer pump signaling that fuel should have been transferring to the header tank; however; the level in the header tank continued to fall. I deselected the automatic mode for the wing transfer pump and turned on the manual switch for the wing transfer pump but the header tank level did not rise. I then turn off the manual mode and turned on the emergency transfer pump which moves fuel from the selected wing tank to the header at a higher rate than the normal wing transfer pumps. The level in the header rose to the max level and I turned off the emergency pump and selected auto mode. About that time the ZZZ Center Controller alerted me that I had climbed through the assigned 15;000 ft altitude (the call came as I was climbing through 15;300). I quickly corrected. I heard the controller ask another aircraft to stop their decent at 16;500. The controller did not order either of us to take evasive action nor di he point out the traffic to me (which I never saw). Shortly after; he gave me a phone number to call at ZZZ Center. Once on the ground at ZZZ I did so and spoke to a supervisor who asked for my phone and certificate numbers. The supervisor said that minimum separation had been violated and that they had to file a report.