As you have suggested, first try using the lap timer via the Arduino Nano's USB port. If the app is allowing you to correctly set the channel frequency and you can see the RSSI spike quite noticeably when you power on/off the VTx this is a good indication SPI is working.
The most common reason it may not be working with Bluetooth is due to the baud rate not being configured to 115200. More details on configuring the Bluetooth modules for PIDflight Lap can be found on the following page: http://wiki.rotoroverflow.com/pidflight:bluetooth
