I’ve ridden Coombe to Watership down many times on the cross (ran it one year, theres a Coombe to Overton 16 miler that’s a really nice run) – you’ll want mud treads and will still likely go on your arse once or twice on the chalk.
I usually do that then bail on road to the Test / Bourne valley home, it’s a nice round trip. Not as well travelled as the Ridgeway by any means.
If you do it look out for the De Havilland memorial stone on Seven Barrow as you slide by.
Going that way its not too rutty actually but it will be wet and sloppy till you climb to Watership Down.