Not inherently. We don't know how it's replicating. Furthermore, this isn't even an an/or situation: the actual virus in the vial of vaccine isn't going to be

Not inherently. We don't know how it's replicating. Furthermore, this isn't even an an/or situation: the actual virus in the vial of vaccine isn't going to be 100% uniform. You could have both virus material that can and can't replicate simultaneously. The extra immune response induced by the subset that can replicate might even make the immune response against the spike protein stronger, even if your theory is correct. Remember that vaccines usually have to use adjuvants to induce a stronger immune response than the active part itself does. An actual infection could do that quite effectively.