A320 CREW ON CLBOUT; DISTR BY TCASII TA TFC; OVERSHOT ASSIGNED ALT.

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APPROX XA13-XA15Z; CLBING ON DEP FROM JFK ENRTE TO PBI. IN THE CLB WE WERE EXPERIENCING MODERATE TURB ASSOCIATED WITH AN AREA OF WX AHEAD. EARLIER IN THE DAY; A COLD FRONT WAS MOVING THROUGH THE NEW YORK AREA CAUSING EXTREME LLWAS WITH LINES OF SEVERE WX. BAROMETRIC PRESSURE WAS EXTREMELY LOW -29.38 IN/HG. BTWN CRI AND WAVEY ON AN ASSIGNMENT FOR DIRECT WAVEY. ATC ASSIGNED HDG 220 DEGS TO INTERCEPT J174; CLB TO FL190. PASSING 17500 FT THE TCASII CALLED 'TFC' (TA). WE VISUALLY ACQUIRED TFC AT 10 O'CLOCK POS; MOVING L TO R IN A R BANKING TURN. OUR ASSIGNED HDG OF 200 DEGS HAD US TURNING INTO THE TFC; WHICH VISUALLY APPEARED TO BE DSNDING TOWARDS OUR ALT AND APPROX 2-3 MI AWAY AND CONVERGING RAPIDLY. OUR ASSIGNED COURSE HAD US TURNING TOWARD THE TRACK OF THE A340; SO I IMMEDIATELY STOPPED MY TURN UNTIL OUR COURSES WERE DIVERGING. WHILE EVADING THE CONFLICTING TFC THE ACFT AUTOPLT CAPTURED FL190 AND DECREASED THE RATE OF CLB FOR THE ALT CAPTURE. CLBING THROUGH FL188 WE PERFORMED THE TRANSITION ALT PROC PER COMPANY SOP. WHEN THE NEW ALTIMETER SETTING WAS SET (29.92) THE INDICATED ALT WAS APCHING FL194. WE IMMEDIATELY DSNDED TO FL190. ATC CALLED AND CONFIRMED WE WERE MAINTAINING FL190.