It does indeed have a bone like texture, but I can't say anything for sure from the photograph I'm afraid, as I would like a lot more detail and ideally to have a look under a microscope. How large is the pebble and would you be able to bring or post it into the museum? Email me on firstname.lastname@example.org to discuss this option further.
In the meantime I'll be interested to see if anyone else can say more from the photograph alone?
I'm not able to bring it to London but I may be able to take it to the NMS stand at the Lyme fossil festival. My camera isn't good enough to get any better resolution pictures sadly. The block is 14*11cm and 7cm deep and there is no sign of the bone sticking out the other side. Here are some blurry pictures of the overall rock;
Bringing it to the Lyme Regis fossil festival sounds like a perfect solution, especially as it is local to the area - there will be lots of expertise there. There will be an NHM fossil tent doing identifications in one of the marques.
I wouldn't have a go at preparing it yourself to be honest, unless you really know what you're doing. I'd recommend taking it to a fossil shop that does fossil preparation and asking them to do it for you. They may be able to explain and show you their equipment and techniques at the same time.
If you actually live near Monmouth Beach there are many fossil shops that do this - The Fossil Workshop in Alice's Bear Shop on Broad Street in Lyme Regis is one of them. But there are fossil shops in most cities (and many fossil-rich towns!)