You need to stay at the correct Hyatt- the Grand is the one connected to your terminal (D).
Regency is a bit of a hassle and you need to get the bus or Skylink round to C and walk through the underground car park.
I have good insurance and would have no qualms about just reserving it myself and then sorting it out with BA/Insurer afterwards.
It's $439 for a runway view tonight.
Remember when you arrive you can use the AA Flagship lounge as an arrivals lounge, just stay airside, if you want a quick bite to eat.