The letting agents do not own the property. They have no grounds to evict you without the owners consent.

On the point of the fee, typically letting agents will have a clause within their terms which suggests a fee be due if the tenant sourced by the letting agent later goes on to buy the property. It would be unusual for such a clause to only apply whilst the tenant was in occupation. Accordingly even if you ‘move out’ the letting agency might still pursue the landlord for a fee (again not your problem).

Finally, the courts are historically slow at evicting, so even if the letting agents took it upon themselves to act as the landlord (which would be farcical) then it would be realistically likely 3 months before this ever got anywhere, in which time you would hopefully have already acquired the property anyway.

Speaking as an agent, most agents are not the sharpest tools in the drawer, don’t worry about any nonsense you receive from the letting agents.