Once the weapons are spawned (and look great) on the server for each Character, we tell Clients to attach the Object references for the weapons that the server spawned:
See in that second image, how the Weapon is spawned with a target Skeletal mesh. That mesh is replicated and gets set up in the construction script (to be fair, really should have worked) but that little bit of construction script wasn’t running on the clients (!?). Run with a [Delay0] after [Event Begin Play] on the weapons; no more racing the server lag (I think).
They were always working, I just couldn’t see them because, you know, no mesh. That should have been the first thing I checked.
I guess it just took documenting the issue to figure it out (it’s been two days q.q)