Reading between the lines, it would appear that the move to a new situation is actually new to all of you -- the father, the kids, and you. So everyone is, perhaps, a little off -balance. Children and cats have in common a difficulty in handling dramatic changes, and both are quick to pick up on tension or uncertainty.
I apologize if I have read your original note wrongly, but I suspect that what is needed is training for the children as well as the cat. If you could engage them in a game (and daddy, too, if you can get him to join), you can do kind of group sensitivity sessions, all joining together in a single room with the cat (an area on a chair or the bed or his special basket can be to one side), and then have some kind of very quiet activity -- board games that everyone can join in, individual activities like working on a computer, reading books, reading a book or story telling to a younger child, coloring books for the kids, etc. The main idea is that all the activities would involve sitting on the floor (except perhaps the computer work), in something akin to a circle. Comfortable cushions or pillows would be a plus.
Part of this activity period could include a snack time (also taken sitting on the floor), which would include a few tablespoons of the cats favorite canned food at the end of the session. You could start by giving him the food near his resting place, and then gradually move it nearer to the human group until it is right in your midst. Throughout this period, no one should attempt to pet the cat. If he comes near, you can speak soft sillyness to him, and if he rubs against someone, they can run a hand gently down his back. But no grabbing, serious petting, hugging, or picking up, and never more than a few seconds at a time. Wait for HIS invitation to touch. He may suddenly find that lying in someone's lap or stretching out along their leg and taking a nap is just what he always wanted. At first he should be able to do this without being touched by your hands, but eventually you might give him very gentle pats from time to time to let him know you love him. A component to this game would be for everyone to pat everyone else very gently on the upper arm, for instance, to show how to do it without disturbing the cat or posing him a threat.
Maybe you could sell it to the family by saying that the cat is very stressed, and that it needs very quiet and structured companionship for a while so that it will get happy and well again. By gradually bringing the cat into the assembled children and adults you will be helping him to adjust to the new situation. But you will also be exercising a strong therapeutic component for the children as well. You will be training them in the patience and quiet that is not only good when they deal with cats, but also good when they interact with adults. They will be learning self-restraint and how to be sensitive to other people's (cats, too) needs and problems. If you and the father are into meditating, you could use a part of this time to have a family meditation, teaching the children a simple mantra method or imaging of appropriate objects. Progress that the children would make in learning patience and quiet would also be very beneficial to the cat (my cats often meditate with me, and it has a very calming effect on them).
If you can make this a regular thing and dress it up with special cookies, cake, mixed nuts, good fruit juice, etc. to add a festive component, and play a little on the sensitivities of the children so they can feel that they are all working to make the poor kitty well and happy, then the rewards could be very great.