pET16b has no leader for secretion ...so we can rule that problem out!
Have you ever checked plasmid stability? ...maybe, since you use a constitutive promoter, there is strong selection pressure towards plasmid free cells or mutations that abolish production of your gene of interest. I would use an inducible system (and stick to the protocol as much as possible!).
Another thing is that they assayed the activity after purification with a Ni-column ...maybe your enzyme is not active in the crude extract? ...or did i got something wrong and you also purified your enzyme?
For expression of this enzyme, Janpanese guy found that the similar amount of protein was expressed with or without IPTG by pET16b vector. It is quite strange, and I don't how to explain this phenomenon. Does it mean that the strong promoter pTrc dosen't influence the expression of protein?
In addition, if purified enzyme has the activity, I think crude enzyme should have activity as well. No inhibition compound was reported.