Short answer to last question: lime, and lime wash (lime plus water). Most mosses need an acidic ph to thrive.
I have no experience with chalk soil, but if it were me - I would begin building soil by:
adding
compost, carbon material + lime on top of what he has now, throw down new seed, do a ph test and keep adding what the new top soil asks for.
Let God and nature worry about the chalk.
Just my two cents, and straight forward way of dealing with things. I'm no expert but I've had a lot of great results fighting moss in the PNW. (could be luck *wink*)