I would not risk adding rain water; sadly it is heavily polluted these days, even in the countryside. There was a post not long ago about some lfs keeper in ?rural Ireland or somewhere who killed off all his fish adding rain water. Much better IMO to let your fish cope with the water as it comes out of the tap.
As you can see above, lots of people manage to keep fish in hard/alkaline water and more fish are used to such conditions these days. OK so you may not be able to breed neon tetras, but a trio of livebearers will fill your tank in no time. Platys, mollies, guppies, swordtails will all be absolutely happy.
My ph was 8 last time I tested (8.4 out of the tap, but I have bogwood which brings it down a little), and my corys still spawn happily. And lots of fish will live comfortably in those waters though they may not spawn.