Removing the barrel trap and installing a P-trap should not necessitate any changes in venting. Venting is determined by the length of the waste arm and it is the same regardless of which type of trap was installed.
To be a bit more specific..The maximum vertical distance from the drain opening to the trap may not exceed 24" except on a washing machine standpipe.
There are tables in the code that list the maximum permissible horizontal length of the waste arm from the Trap weir(actual point of discharge in the trap) to the vent. The actual mathematical method of computing the length is by deviding the diameter of the waate arm line by the fraction of an inch pitch required. A tub requires an 1.5" trap, which is required to have a 1/4" per ft. pitch, thus the maximun lengt of the waste arm is 1.5/.25= 6ft. The code does allow the waste arm diamter to be increased one nominal trade size larger than the trap it serves therefore you could increase the waste arm to 2" which would yeild 2"/.25= 8ft.
Regardless of what type of trap is installed, if the total developed length of the waste arm exceeds 8' you would be required to install an auxillary vent for the tub.
The Uniform Plumbing code (UPC) has derated the mathematical maximums by about 50%. They list the maximum for an 1.5" line at 3ft 6in and a 2" line @ 5ft. (Check your local code authority to see which code applies in your jurisdiction)