Just some quick remarks & hints:
Bluetooth low energy connections have interfame spacings of 150us. So having a connection with the LimeSDR is not possible unless you implement the controller on the FPGA (for modem) & FX3 or nios cpu inside the FPGA for logic. Worst case the end of the last packet before a frequency hop is 150us before the next packet. However XTRX which uses pci express and is also lime based https://www.crowdsupply.com/fairwaves/xtrx may support 150us latency depending on the host cpu & os, and should have the device bandwidth for 80mhz.
However most of the time the channel will be changed every x milliseconds (where x is > 7.5, typically 30, 50 or a few hundred). In these cases it might be possible to follow the connection if you are able to sniff the connection phase and determine the connection parameters.
It is possible somewhat to sample the whole 2.4 ghz band. I have been able to see the waterfall of 2000-2480 mhz on sdrangel, but it has some artifacts (high intensity areas) and dropped frames. If you make a dedicated application you may be able to solve this, for example by reducing the number of bits per sample.
I guess making the software required for a bluetooth sniffer using limesdr will require a significant amount of time especially if you want something stable and user friendly, so if you are not into this for the challenge (and want something usable) i would suggest pursuing other options.
Commercial products that does what you describe are markeded by Teledyne LeCroy and Ellisys among others, but they are much more expensive than the adafruit sniffer.