Hmmm. Google maps give a present day journey time of three hours via the M3 between Bournemouth & Hastings. The stagecoaches prevalent in 1887 would probably average little more than 4mph (At this time road locomotives were limited to a maximum speed of 4mph in open country and 2mph in built up areas, so it seems reasonable to assume that average speeds for horse drawn traffic would be similarly low) on such a journey, and there would definitely be a requirement for a change of horses at least once. Therefore Id suggest, although I do not know for sure, that travelling by train via London would be faster than travelling along the South Coast by coach.
Id think that the favoured route from London to Hastings (after travelling into London from Bournemouth on the London & South Western main line through Southampton & Basingstoke), would be on the South Eastern Railway through Seven Oaks, Tonbridge & Tunbridge Wells. I dont know what the journey time would be, but I do know that a comparable London to south coast journey with the neighbouring London, Brighton & South Coast Railway from London to Brighton could be done in 1 hour 50 minutes in 1910 (the earliest date I have any sort of info for), by the fastest express trains on the route. The London & South Western Railway main line from Bournemouth would have been quite quick for a railway of the day, and the LSWR were quick to introduce smoother riding bogie coaches from the 1880s onwards to replace the four and six wheel carriages prevalent on passenger trains at the time.
I think it would be reasonable to assert that an eight hour journey time from Bournemouth to Hastings via London would have been possible in 1887, dependant of course on train connections, as direct trains between towns were very much fewer in number then than they are now. There would also be a station transfer across Victorian South London needed, which may not have been to everyone's liking on a journey such as this.
Im surprised no-one else on this forum has been able to provide any more specific information. My knowledge of railways is much more focused on northern England than the southern end of the country and it would be good to have a further opinion.