RE destination - I have no clue on the Hyatt scheme and what that will get you, however I would suggest you start with this side. You have plenty of miles for a business (or first) flight so I would look at what you can get hotel wise first and then plan from there.
Generally the advice here is to use miles plus cash but you want to save cash and have plenty of miles so miles only is probably the way you'll want to go.
Taxes and fees are higher for business so if you really are looking to minimise cash spend you'll need to go economy - but that really isn't good value for miles and IMO the difference in tax is worth it so if you can raise the cash try to go business. Alternatively if you don't mind a little inconveinice you can look at positioning yourself outside the UK where tax is lower, maybe even have a few days in a European city so its not just an extra connection?