I doubt there are moons around the tidally-locked planets. Over large amounts of time, moons end up tidally locked to planets, but if the planet is already tidally locked then the the only places that the moon could still be gravitationally bound the the planet and still be tidally locked would be the Lagrange points which aren't especially stable (or close to the planet).
The Earth is not tidally locked to the Moon which is tidally locked to it.
The complications of the Sun's towards-a-tidal-lock-upon-Earth influence also affecting the lunar orbit
isn't inconsequential, but I don't think there's any technical reason why a moon can't be orbiting a planet, locked to it, whilst the planet is orbiting the star and locked to said star.
(Note that L1 and L2 (also L3) are unstable points of equilibria, so I would not expect a natural 'perma-conjuction/opposition' to form up. L4/5 are loosely stable, but expect a 'kidney-bean' orbit without artificial efforts to place the moons into the ioscelesal third point, effectively ±60° off in the same orbit.
All this vastly complicated by the masses of the moons (a factor not usually significant with spacecraft, in their perturbation of the barycentre of all three objects), and of course the additional planets (and moons) co-orbiting in dissimilar resonances complicate things even further. If there
are L4/5 moons, in principle, then I expect them to occasionally 'defect' - if not get thrown away beyond
any planet's momentary 'ownership'. Seven significant planets in narrow orbits probably already jostle around and 'migrate', their ancillary satellites little more than chaff on the stellar winds, but I've yet to look at the specifics so far discovered about this system. And smarter minds than mine will have already shoved all available data through countless analyses that are beyond even my own imagination, never mind expertise.)
It would be tricky to get non-mutual/cascading locking, but less tricky than any three-body locking [well, two-body and the star yet to be dragged to a rotation matching its subordinate partners... that last step will probably take longer than the system has 'life' for]. I think, more likely, would be that chaotic degeneration of any star-planet-moon system, as described, before the planet locks to star. But it would benefit from some many simulations of wide ranging and contrived setups, to be sure.