Concrete fasteners - problem
Try using 5/16 screws. If Tapcons don't come in that size, you can use stainless wood screws. Steel screws will rust. Although this is wood trick, rubbing the screw in bar soap helped me once. If you can, should remove the threshold and vacuum out the hole. Hand screw only.
You could try this. Protect the threshold hole with tape. Squeeze in epoxy, one that looks like 2 part syringe, coat the tapcon with same epoxy, set into the hole and weigh the screw down. Masonary epoxy may be more logical choice, but applicators wouldn't fit into the holes.