And the answer is, yes, a cork underlayment substantially reduces impact noise transmission between floors even with fasteners connecting oak plank to subfloor. Footfalls are muffled to the point that you have to listen for them. I couldn't find a definitive answer beyond "cork is good but don't nail the floor." So I split the difference, using a high performance polyurethane glue with a bare minimum of fasteners. There is not much choice with plank because you need to pull the board into the glue, cork or no cork. My spacing was 12 inches on center. The glue was bostik's best. I used screws to stay out of the glue, which is pretty close to permanent once it sets. {It is on Warmboard's list of acceptable glues over radiant.) So far so good. There isn't a squeak anywhere. So much for the good news. Turned out to be very labor intensive. You are gluing three floors: half inch cork to Advantech; quarter-inch underlayment to cork; oak plank to plywood with transfer fins in between. A better answer is to float engineered hardwood over half inch cork with the radiant under the subfloor, or with 3/8s pex if it fits in the cork. My plank came from trees on the site that had to go.

Have you determined if you can get any heat transfer through this emitter? Cork is a great heat transfer insulator and is used on spacecraft heat shields to keep them from burning up on re-entry. Hopefully the cork layer is below the heat transfer plates. A finished wood floor also reduces heat transfer, but this can usually be addressed with a higher supply temperature.

The transfer plates are in direct contact with oak plank. (Advantech>cork>plywood>transfer plates>hardwood) I've found values of R 1.56 for half inch cork plus R0.25 for plywood underlayment and R0.68 for 3/4 hardwood. So the resistance down is 2.6 times the resistance up. Heat will take longer to get there through the oak but I'll need a chilly morning before determining if I need higher temps. But you are right that staple up radiant under cork is a nonstarter. Floating or gluing engineered over cork is done all the time in condos using thinner underlayments than half-inch. Dunno if anyone has tried half inch and radiant. Seems to me, flooring people don't know radiant and radiant people don't know flooring, and both have an overabundance of caution.

What I know for sure: Nailing through cork may degrade sound control, but a modest number through half inch cork is still a major improvement.

I installed a floating engineered cork over 1/2" plywood covering the original 3/4" T&G pine planking with the original 'C' fin extruded aluminum plates and 1/2" PEX, with R-19 fiberglass for our second floor bath. All works perfectly with a SWT of 120°F at design conditions (-13°F).

Response time is eliminated with by enlisting weather responsive controls and foamed walls.