Thank you, all.
Sorry I dropped the ball there for a few days. I received several interesting off-line answers in addition.
I think I should also try to clarify. First off, for the (some) botanists - in entomology there is much less of a tradition of "creating duplicates" (of purportedly the same individual..thinking about branches of an oak tree here). Insect specimens overwhelmingly remain and travel "entire" (even following dissection). I hope that distinction is fair enough to most.
Here is the conflict, as simple as I can state it. There is an institution that the specimens ultimately belong to, and that loans them out to a researcher. Then there is a researcher, not affiliated with the institution, who right now has resources and arguably needs to "publish" the specimens via iDigBio, GBIF, etc. (as well as other outlets such as a research journal).
Let's assume that the owning institution just really does not have the resources right now. Not even to put a locally unique specimen identifier on it (or, it does that, but there is no digital counterpart). And the researcher does. Beyond writing a kind, explanatory e-mail, and figuring things out (idiosyncratically), is there some more widespread accepted practice for resolving this conflict? Answering "I use this or that portal that I happen to have access to and which does it for me", is not really a generally applicable answer, right?
If not, should we as a community (in our most hopeful moments, anyway) consider creating one or more that are very open for contributions? Something like an open portal for digitizing and iDigBio-/GBIF-publishing research-relevant specimens of/for owner institutions that "just can't right now, sorry".