Let us know how it goes...
When I bottle conditioned, I was never able to really get the carbonation consistent from one bottle to the next using the batch priming method.
I don't know about cider, but with beer, you don't want to stir much due to oxygen exposure concerns, and you're already compromised using the bottling bucket here anyway.
Shortly before I went to kegging, I started using the "fizz drops", sold under a number of brands; they are sugar tablets that are a consistent dose in each tab, and depending on your target CO2 volumes, you drop X number of tabs in each bottle.
This gives consistent carbonation, and you should also be able to bottle straight from the fermenter and skip the oxygenating bottling bucket step.
The tabs are a little more expensive than batch priming for sure, but in my opinion, worth it... ymmv.