You detect the chirp, and subtract it from the mixture. The remainder is your signal of interest (plus other sources of noise which you don't mention). Then you calculate the power of each component, and calculate the ratio. That's SNR.
The challenge is usually detecting the chirp, but that can be done, depending on the level of information you have about it (chirp rate, phase of the chirp, beginning of the chirp, bandwidth, amplitude) using simple techniques like a correlation with a prototype chirp, or "blinder" methods like letting a PLL run on the signal and watching when it locks.