Microwave-dressed Rydberg atoms in pairs of states that are optically accessible and engineered so that the frequency of the transition between them is tunable while exhibiting a low sensitivity to stray electric fields have been interfaced with a superconducting coplanar waveguide (CPW) microwave resonator. This was achieved with helium atoms in triplet Rydberg states with principal quantum numbers $n=55$ and 56. The atoms were dressed with two strong microwave fields—a control field and a differential polarizability nulling field. The control field allowed the atoms to be coupled to the resonator using the two-color two-photon transition between the equal parity $1s55s{}^{3}S_1\equiv |55s\rangle$ and $1s56s{}^{3}S_1\equiv |56s\rangle$ Rydberg states, with one photon supplied by it and the other provided by the resonator. The nulling field was detuned from the field-free $1s55s{}^{3}S_1\to 1s55p{}^{3}P_J\equiv |55p\rangle$ transition to admix $|55p\rangle$ character into the $|55s\rangle$ state and eliminate the differential static electric dipole polarizability of the $|55s\rangle$ and $|56s\rangle$ states. The experiments were performed with atoms located $\sim 300\mu \mathrm{m}$ above a $\lambda /4$ niobium nitride superconducting CPW resonator operated at temperatures between 3.59 and $3.73\mathrm{K}$. The measured ac and dc Stark shifts of the Rydberg states at this atom-circuit interface in the presence of the dressing fields are in good quantitative agreement with the results of Floquet calculations. The differential polarizability nulling fields used in the experiments allowed the difference in the Stark shifts of the $|55s\rangle$ and $|56s\rangle$ states over the range of dc fields offset from the residual stray field by $\pm 65$ mV/cm to be reduced from $2\pi \times 2.1$ MHz to $2\pi \times 125$ kHz.