CLB OVERSHOOT IN TSTM CAUSES LTSS.

Narrative

CLBING OUT OF JAX ON 360 DEG HDG CLBING TO 9000 FT ASSIGNED. CAPT ASKED 30 DEG L TURN TO AVOID A TSTM. JAX DEP CTL DENIED REQUEST. APPROX 2 MINS LATER WE ENTERED THE BUILDING TSTM AT APPROX THE SAME TIME THE CAPT BEGAN HIS LEVER OFF AT 9000 FT MSL. WE ENCOUNTERED HVY RAIN AND AN UPDRAFT THAT CAUSED OUR ALT TO INCREASE TO 9500 FT MSL. WE EXITED THE TSTM AFTER 20-30 SECONDS TIME. THE JAX CTLR ISSUED EVASIVE MANEUVERS FOR US AND CONFLICTING TFC. I HAD THE TFC IN SIGHT. THIS ALL COULD HAVE BEEN AVOIDED HAD WE BEEN GIVEN OUR AVOIDANCE VECTOR AS REQUESTED. SUPPLEMENTAL INFO FROM ACN 282841: REQUEST A 330 DEG HDG TO AVOID GOING INTO THE ONLY TSTM IN THE AREA.THE ACFT STARTED ASCENDING AT APPROX 2000 FPM. THE ACFT STARTED TO DSND AT APPROX 9500 FT BUT NOT BEFORE THE CTLR TOLD US TO HOLD 9000 FT BECAUSE WE HAD TFC AT 10000 FT. THE CTLR TURNED OUR TFC L; WE WERE SEPARATED BY ABOUT 500 FT VERT AND 5 MI HORIZ. THE CTLR MADE A ROUTINE SIT DANGEROUS. I SHOULD HAVE BEEN MORE INSISTENT ON NOT GOING INTO THE WX. NO ONE ON BOARD WAS HURT.