Is it normal for wire wheels to slowly lose air around the hub?

The shop guy is half right. Wire wheels absolutely are famous for slow leaks like yours, but it's not the splined hub. The air in your tire never touches the hub - the bead sits on the wheel's own rim, and the splines just drive the wheel. There's no path from the tire cavity to the adapter, so that leak theory doesn't hold up physically. What is real is leaking thru the spoke nipples. On a wire wheel, the spokes pass right through the rim and the nipples end up inside the tire's air chamber. Tubeless tires on those rims leak past the threads and the little holes, slowly, exactly like you're describing - a few psi over days, one wheel worse than the others, nothing you can find at the bead or valve. The leak may also only show once the tire is loaded and flexing, which is why a shop check on a static wheel sometimes misses it.

Easy way to confirm before you pull anything: mix up some soapy water and paint it over the whole rim with the tire at pressure, including every nipple. If you've got a bead of bubbles forming at the spokes, that's your answer. If it's clean, then it's worth rechecking the bead with the wheel on the car and loaded, since a slight flex can open a gap that only shows under weight. If it does turn out to be the nipples, the standard fix is to run an inner tube. Lots of people do this on wire wheels anyway and never think about it again, and it's a cheap job while the tire's alredy off. The alternative is sealing each nipple from the inside with a suitable sealant or a bead of silicone, but that's fiddly and tubes are more reliable long term. Some tire shops won't mount tubeless on wire wheels at all for this reason.

One other thing worth checking while you're at it: a two-year restomod sometimes has a wheel that's been sat on a shelf and taken a knock at some point, and a slightly buckled wire rim will weep at the bead even with everything else perfect. If the bubbles point you to one spot on the rim rather than the spokes, get that wheel trued before chasing anything else.

Went through this exact headache on my TR4 years ago. Kept chasing that one slow wheel with soapy water and never found a clean answer, so I just had tubes put in all four next time the tires were due. Haven't thought about it since, and that was maybe a decade ago. One tip from that job, tell the shop you want tubes fitted and make sure whoever does it knows wire wheels, a lazy mounting job will pinch a tube and you'll be back where you started. Also keep a small 12v compressor in the boot during the break-in period just in case a nipple lets one down faster than you expect.

Ran tubes on my MGB's wires for years and the thing that actually fixed my slow drops was proper cloth rim tape. The first lot of tubes kept needing air every fortnight becasue whoever mounted them had used a thin plastic strip, and it wandered sideways so a spoke end chafed a pinhole in the tube. Fresh tape and the wheels held air for months at a stretch after that. Side benefit nobody thinks about with tubes, you can let a tire down for a look without the bead popping off the rim, which is a real pain on old wire wheels. Makes checking pressures and fiddling about at home much less drama.

Update: took the wheel off and dunked the soapy water around the bead like one of you suggested, and I can see tiny bubbles coming up through the spoke holes on the inside of the rim. So it does look like the rim itself is the problem, not the hub. Before I go the tube route, does the cloth rim tape trick apply even without tubes, or is that only for keeping spoke ends off the tube?