Hi Branwen
A few miles away from Bournmouth, but we take regular weekend breaks in Lyme Regis and stay at a lovely hotel called Hotel Alexandra, Pound Street, Lyme Regis.
The hotel is very dog (and kid) friendly, very close to the sea, centre of Lyme Regis, and the hotel itself is charming.
If you want more details, let me know and I'll send you the link. As a matter of fact we're off there next week, and looking forward to letting Molly have her first swim in the sea (she loves rivers and streams, so it'll be interesting to see what she makes of waves!!!).
Best regards
Mark