In reply to summo:
Just about every national organisation in the world such as the bmc have done rope damage tests, plus other independent non rope makers. There must be data going back 20 plus years since ropemen, grigris and other more aggressive devices first entered the market.
Is the data from these tests (BMC technical committee etc.) available to the general public? So far the most interesting source I've found is the Black Diamond QC lab https://www.blackdiamondequipment.com/en_US/experience-home?fid=qclab - these are the sorts of tests that I like because they back up a lot of the widely accepted practices with evidence (classic example being double fishermans vs double overhand for tying ropes together when abseiling). Unfortunately they often only measure the test cases once, so there is still significant uncertainty in the results.
Within the context of the original question and the available equipment, I agree with the "just don't do it" approach, making sure to be using an undamaged section of rope instead for any weight bearing. However, if there was a device (or sufficient proof for an existing device) that allowed "safe enough" travel on a compromised rope, it would be interesting to hear about it because it would work in a wider range of situations, eg. ascending a single strand of rope tied off at the top (this happens fairly frequently when climbing on sea cliffs, and it is conceivable that the rope gets damaged due to lack of rope protectors and using a dynamic rope). I think that data on testing of existing equipment beyond their UIAA specified purposes is always relevant, as it allows us to understand the subtleties of different designs and use it to our advantage or avoid gear-specific pitfalls. If there is sufficient evidence that a novel technique involving a particular piece of equipment is faster than an existing technique and safe enough to be acceptable in a pinch, why not adopt it?
Also, if someone were to ask the question "what happens when I try to climb a rope with a damaged sheath using a classic prussik/tibloc/ropeman?" and nobody knows for certain because it hasn't been attempted, wouldn't it be interesting to find out?
You never know, the next time you do a bouncy abseil down a sharp-edged sea cliff, throw your climbing rope and rack in the sea to find yourself stranded with nothing but 30 different rope clamps/ascenders, you might just be grateful of some prior experiments to help you choose your best option!