Hmmm... hmmmm... so riding in the country was an enjoyable experience. I am not surprised. Our first tandem was very similar to your Burley except it was a flat-bar model. 700C wheels, disc brakes so intended for fast club riding, but flat-bar cockpit. When I converted it to drop bar for even more speed after a couple of seasons I gave up 3" of leverage on each side of the bars just like that. Riding in Brooklyn, NY and lower Manhattan were NOT a lot of fun. So we got another flat bar model and left it stock for town cruising, errands, etc. and used the drop bar sport ride for club rides.
You may already be doing this, but I would confine rides on the Duet to the park or otherwise off the main grid for the time being. Rides need to be fairly often so the Captain can build confidence. Riding the tandem solo as captain is a good exercise as the solo tandem has some of the flop of the loaded rig (no offense) but there is none of the risk of stoker injury to cause concern. Slightly panicked, death grip on the bars describes my approach even after 12ish years of nearly daily commuting, lol. Are you sure your Captain isn't getting more comfortable? Has he voiced it or are you inferring from non-verbal cues?
I would advise not altering the shifter locations. I would not even give drag brake control to my stoker. IMO the Captain should be in direct control of shifting, braking, etc. Our bell is, however, on her bars. When I need to 'shout out' to someone, I have to relay a command through her. Shows where I rate bells on the importance scale.
How much riding together do the two of you do now? Are your commutes done in tandem (as it were) or do you head off in different directions? I ask because it might be that riding a tandem is just not something for which your Captain is suited. We belong to a fairly large tandem club and most teams have a clear difference in riding experience, power output, etc. between Captain and Stoker. In our case my stoker is blind. If she were sighted it is unlikely that we would be on a tandem. We would still enjoy riding together, but it would definitely be on separate bikes.