As ninfan said you’d need to gather evidence of 20 years use by horses (as BW’s can’t be claimed for by use on bike) in order to put in a claim, but seeing as it’s only relatively new the claim won’t pass the first hurdle unfortunately.
So, I’d get in touch with sustrans to see if they originally contacted the landowner/s (of the bit further along which isn’t a cycle route) as they’d have worked as far along the old canal as possible and only stopped when they reached a ‘no’.
If they hadn’t asked them (or if they were originally turned down, its been long enough for them to ask again, it’s worth a shot TBH) then ask if they could do so now as you’d really like it extended for safety reasons. Maybe get a list of locals who’d like it opened so you can say ‘look, these 50 people would like to see it opened too’. If they umm and ahh then say ok, I’ll ask the landowner yourself to see if they will allow permissive access on bike / horseback.
As to who pays for replacing the stiles with gates, assuming the landowner is OK with it, then get back in touch with sustrans and your local Rights of Way department with a proposal. Get lengths / numbers of stiles / gates / etc so they at least can work with something solid 🙂