Use @unchecked Sendable. If the only thing that doesn't conform is DispatchQueue, then you can tell the compiler to shut up: you know that DispatchQueue is actually thread-safe, so you can just mark your type appropriately.
That is a strong word that infers moral obligation. I would say that; DispatchQueue safely meets the requirements of Sendable except one set of caveats; DispatchQueue.sync. If you avoid that and similar functions then the type is reasonable to use in an actor and in conjunction with existing systems.
While Swift concurrency does strongly discourage the use of thread-blocking patterns like DispatchQueue.sync, the method itself doesn't violate any sendability restrictions. And as long as the queue isn't backed by a width-limited queue, you won't get exhaustion deadlocks. You can, of course, get other kinds of deadlock if you call it while holding some other thread-exclusive resource. @rokhinip may be able to correct me here.