AN ACR BE02 FLC RECEIVES A TA FROM THE TWR CTLR AND THEIR TCASII AND PROPERLY REACT.

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ORIGINAL CLRNC 310 DEG HDG AND 1600 FT. OUR ACFT WAS CLRED FOR TKOF ON RWY 36R AT TPA WITH AN AMENDED HDG OF 'RWY HDG.' IMMEDIATELY AFTER TKOF; THE TWR POINTED OUT TFC NEAR US AND OUR ALT AND INFORMED US THAT HE WOULD TURN US AS SOON AS HE GOT SOME ALT FROM US. THE FO CALLED THE TFC 'IN SIGHT' NEAR US; THEN LOST CONTACT. THE TCASII INDICATED THE ACFT WAS CLOSING IN AND THE ALARM WENT OFF. I STARTED CLBING TO AVOID THE TFC. I STOPPED AT 2200 FT MSL THEN RETURNED TO 2000 FT MSL AND INFORMED DEP (AS WE WERE SWITCHED OVER BY TWR) OF OUR ALT. THE FO TOLD DEP WE WERE AT 2000 FT AND THAT TWR HAD TOLD US TO CLB TO AVOID TFC. DEP CTL HESITATED; BUT DID NOT QUESTION FURTHER. OUR ASSIGNED ALT HAD BEEN 1600 FT MSL; BUT DUE TO THE ENCROACHING TFC AND THE INFERENCE BY TWR THAT HE NEEDED A CLB; I BREACHED THE ALT. RATHER THAN RETURNING TO 1600 FT; DUE TO THE CONFUSION; I THOUGHT IT BETTER JUST TO INFORM DEP OF OUR ALT. WE WERE PROMPTLY CLRED TO A HIGHER ALT AND NO FURTHER DISCUSSION TOOK PLACE. AFTER THE FO'S INITIAL CONTACT WITH THE OTHER TFC AND SUBSEQUENT LOSS OF IT; IT WAS NOT SEEN AGAIN.