The receiver is performing the RAIM function all the time.
The prediction function just predicts whether or not RAIM will be available at a particular point at a particular time.
The prediction function in the GPS uses the GPS's onboard almanac to do the prediction and does not take into account any space vehicles that may have been taken out of service. The most accurate RAIM prediction is one that uses up to date data from the US Department of Defence. So, yes it is more accurate to do a RAIM/FDE prediction using the Air Services service.
The FDE programme that I am familiar with requires you to enter, which if any, space vehicles are out of service before it does the prediction. The data on the space vehicles can be got from the US coastguard site.
GPS Constellation Active Nanu Status
I don't see the the point of doing a RAIM prediction just prior to the approach. I've never heard of this requirement. Why would there be any change in the prediction from the time you did the prediction prior to your departure.