How can I experimentally find the bandwidth of my PLL?
A step response test is an easy way to determine the bandwidth. Sum a small step  into the control voltage of your oscillator (VCO or NCO), and measure the 90% to 10% fall time of the corrected response at the output of the loop filter as shown in this block diagram. Note that the loop will respond in such a way to completely cancel the injected offset, but can only respond at a rate within its bandwidth. Given this is in an FPGA implementation, this would be an easy test feature to have permanently included, especially valuable if this is a mixed signal system with external analog components that may vary from unit to unit or over temperature.

The bandwidth is given by the following approximation (this is accurate for a first order loop but will provide a reasonable estimate for higher order loops as well):
$$BW = \frac{0.35}{t} \space\space\space\space \text{(first order system)}$$
Where $BW$ is the 3 dB bandwidth in Hz and $t$ is the 10%/90% rise or fall time. This equation comes from solving the first order response equation given by $y(t) = e^{-t/\tau}$ where $\tau$ is the reciprocal of the bandwidth in radians/sec. For second order loops with a typical damping factor of 0.7 this relationship is closer to:
$$BW = \frac{0.33}{t}\space\space\space\space \text{(second order system, damping factor = 0.7)}$$
You could use this test for further loop forensics such as damping factor, natural frequency and a more accurate estimate of the bandwidth through observation of the overshoot and ringing in the corrected response.
The step should be as small as possible to stay within the tracking range of the loop, but large enough to make a reasonable measurement of the response time.
Below demonstrates the accuracy of using this first order approximation on a second order system, showing the step response and frequency response of a generalized 2nd order system with a natural frequency of 1KHz where the damping factor was varied between .5 and 1.0. The dashed red lines indicate the 10% to 90% crossings for rise time and the -3 dB BW crossing for the frequency response.
The following table shows the resulting accuracy of the approximation versus actual 3 dB bandwidth for each of these cases.