Read up on the Fourier transform.
If you want a short pulse, you need a wide bandwidth.
If you need a narrow bandwidth, you need a long pulse.
This is true for baseband (step response, pulses, general signals: a center frequency of 0Hz), as well as RF (the same sorts of signals, around a carrier; a center frequency higher than the signal bandwidth). So, a wavelet with narrow bandwidth must ring for a long time.
If you need short impulse response, consider a Bessel type bandpass. Don't be surprised if the frequency response turns out horrible, though.
Likewise, we can see what your above measurement has produced: notice the tone burst starts on a sharp corner -- it's differentiated through the network's series capacitance, not "filtered" at all. (One property of the crystal ladder filter is, the asymptotic response -- the attenuation at frequencies far from Fc -- is not very good, typically 20dB per crystal used, with peaks and dips at various spurious frequencies where the crystals have secondary resonances. They are best used when combined with a fairly sharp, modest order, LC filter.) This is also why the amplitude is so small: the tone burst length needs to be on the order of 1/BW to see the output ring up to its full output. The ringing seen afterwards is the impulse response (since such a short burst is near enough an impulse, being much shorter than 1/BW), which will have the same form (decay over time) for a longer burst, as is seen here.
Finally, your scope measurement will need to reflect the signal as well. I suggest triggering on the generator's GATE output signal (or AUX or whatever it's called), setting horizontal timebase on the order of 1/BW, and setting acquisition to peak detect mode. (An analog scope will read the RF envelope just fine; a newer scope may not need peak detect if it has deep memory, samples faster than the center frequency, and displays the waveform with antialiasing.)
Cheers
Tim