bandpass waveform detector?
Very interesting problem.

Your pulse locations can be found by using a sliding window RMS measurement.  Once found, I would recommend centering a DFT frame on the pulse and applying a VonHann window (broadens narrow peaks and narrows broad peaks), then take the FFT.  Looking at the data, you should get a pretty distinct and clear profile.

Your backgound noise may contain some small peaks too, so you may want to analyze a few "quiet periods" to determine what they are so you can discount them in your evaluation.

Precise frequency determination shouldn't be necessary (or even possible) so a simple estimator, like Jacobsen's or MacLeod's, should suffice.

Like wise, a rough amplitude estimate should also be adequate for each peak.  (See my answer here for phase and amplitude estimation: FFT Phase interpretation of input signal with non-integer number of cycles in FFT window)  Don't use the window if you are going to calculate these.

If you know the frequencies ahead of time (or want to do it twice), selecting a DFT bin that has your frequencies near bins will improve these estimation methods considerably.