Do I need LEV testing if my system is relatively new?

You do, indeed. At least once every 14 months, or more frequently depending on the procedure, even a new system needs to be evaluated. The explanation is straightforward: a brand-new system may still be improperly installed, configured, or just unsuitable for the hazard it is intended to control.
The first statutory test establishes a baseline, demonstrating that the system functions correctly right away. You can not demonstrate to inspectors that it is performing its duties without such. Comparable to a car's MOT, the legislation still requires a check regardless of how new the vehicle is.