The more recently introduced inerter fictitiously increases the mass of the system to which it is connected, creating a mass amplification effect. This property makes it particularly attractive for use in mass-dependent devices employed in structural vibration control, i.e., the prevention and mitigation of vibrations. Conceptually, the inerter can be viewed as a two-terminal device, where its internal force is directly proportional to the relative acceleration between the two terminals. The constant of proportionality, called inertance, is the apparent mass generated, which can be notably large. In practical applications, it is effectively utilized in combination with springs and dashpot dampers, giving rise to Tuned Inerter Dampers as an alternative to traditional Tuned Mass Dampers (TMDs). Among the various prototypes of inerters, fluid inerters stand out as simpler to design and develop, with less pronounced parasitic effects compared to their mechanical counterparts. However, their inherent nature makes them susceptible to nonlinearities, such as fluid viscous shear friction, pressure drops, and other tribological effects. To mitigate these undesirable nonlinearities, a proposed solution is to connect the fluid inerter to the structure through a spring-dashpot damper element, referred to as flexible connection. To evaluate the practical feasibility of this type of connection, a novel experimental test is presented, focusing on a fluid inerter used for vibration control in a cross-laminated timber panel subjected to vertical vibrations. In addition, the control performance of the fluid inerter is compared with that of a conventional TMD, providing insight into its efficiency in controlling multi-modal structures. Finally, optimization procedures are employed to identify the connection parameters of both the TMD and the inerter, aiming to improve their performance in effectively controlling the structure.