Will the SiriusXM SXAR1 be able to record data when the iPad is off or the app is backgrounded?

Because the SXAR1 connects via Bluetooth it is able to maintain the connection to the iPad or iPhone and ForeFlight Mobile even when the device is asleep or ForeFlight is put in the background.

So, as long as you don't turn the iPad OFF or completely quit ForeFlight Mobile, the SXAR1 will keep feeding radar and other weather data to ForeFlight Mobile while it's in the background and/or the iPad screen is off.  When you re-open ForeFlight Mobile, it should have all of the past radar data saved, so the animation will work right away.