I'm not in NHS Trust but will give you my tuppence worth from (similar?) LA perspective.

From a CDM perspective, you will remain the Client. From a Control of Asbestos Regs perspective, you will remain the duty holder. Accountability and duty to ensure what should be done is being done, will always ultimately come back to NHS.
You can explore competency at PQQ and tender stage to some extent, however to ensure you continue to comply with the law, and for continuing assurance, you will require robust arrangements, resources, monitoring and audit processes.
Some of this monitoring can be placed in the contract terms, but would still need to be verified by audit.
Bottom line: where you engage any contractor, you need to retain resources in-house to ensure continuing competency and compliance.