It is the duty of the Responsible Person to 'provide measures in relation to the means for fighting fires in the premises'. This does not specifically say that the Responsible Person has to provide and service the extinguishes themselves.
As I see it the situation you are describing has the possibility of three Responsible Persons. The Landlord/owner of the building and the two Trusts, each of whom has a responsibility to provide fire fighting equipment. How this is achieved is up to them. They can either provide the extinguishers in their bit or the landlord could provide all the firefighting equipment as part of the leasing agreement, or the Trusts could get together and agree to obtain the extinguishers from a single source and share the costs.
The FSO tends to tell you what you need to do but does not tell you how to do it. This is left to the guidance and standards.