It sounds like if at that point (after throwing everything at it including DTT and papain) you don't have the rare antisera/cells and reagents to identify the antibody then you are going to have to send it out anyways and the reference lab will probably end up repeating most of that testing in order to ID the reactivity on their end. You are right, you have to consider that implementing trypsin is a big project with SOP revisions, may involve added costs, is known to have stability issues, and how often do you expect you will really need it and keep your staff competent? Will it really have any added benefit to your testing?
From personal experience in a smaller reference lab with limited rare reagents as well as in a larger IRL, I would say it sounds like you may want to wait on the trypsin until you have a better inventory of rare reagents to aid in completing antibody ID?