In my experience there is no difference between the pickle characteristics of 304 and 304L.
I need some more info. Is the annealing of your 304 L carried out in an lectric or gas furnace? Is the surface oxidised when it exits the annealing furnace? If that is the case, you removed the passive layer of Chrome oxide while annealing and why the pickling process is that much faster.
In the first instance with plain 304, the steel structure, except the weld, were still passified from the pickling process. The mixed acid thus attacked the oxidation in the vacinity of the weld, but the general steel surface was only mnimally attacked due to the passivating oxide layer.
I am not a metallurgist, but do you really need the second annealing step?