The shielded pair resonator method is a useful tool in the measurement of accelerator components, such as the beam screens used in the Large Hadron Collider (LHC), the High-Luminosity (HL) LHC, or future accelerators. It can measure the resistive losses at several frequency points by separating the resistive losses on the sample from other sources of losses. We built a new resonator to be inserted into a superconducting dipole magnet (peak magnetic field of 9.5 T) and to measure the surface resistance of beam screens, such as LHC beam screens coated with amorphous carbon (a-C). The device can measure surface resistance at any temperature between 4.2 K and 300 K, in the frequency range of 400 MHz to 1600 MHz. We conducted the first surface resistance measurements of two a-C coated beam screens at 4.2 K and showed that the 200 nm to 400 nm titanium underlayer plus 50 nm a-C only has a limited effect on the surface resistance. This first result supports the choice of this coating as baseline for the HL-LHC triplets magnets upgrade. The resonator will have an important role in the characterization of next-generation beam screens, such as a beam screen with laser-engineered surface structure (LESS). Further measurements of the LHC beam screen in the presence of magnetic fields up to 9.5 T and throughout the full temperature range are going to be reported separately.