Imposing sparse structure in Gabor space
In my experience, you do not necessarily need an orthogonal dictionary, see for example Compressed Sensing with Coherent and Redundant Dictionaries (DOI: 10.1016/j.acha.2010.10.002).

The paper Quantifying the performance of compressive
sensing on scalp EEG signals (DOI: 10.1109/ISABEL.2010.5702814) seems to be an example of the use of Gabor dictionaries (in EEG) in compressed sensing. I would just go ahead and try it.

If the chirp you are looking for can have a fairly fixed location in the Gabor domain, it might be possible to use compressive detection or classification instead of actual reconstruction. This is explained in Signal Processing With Compressive Measurements (10.1109/JSTSP.2009.2039178. You can see an example of how we have used it for symbol detection in IEEE 802.15.4 communication.