from your description it sounds like the behavioral problems at home have somewhat already eased down. This is very good news and it seems that she is able to respond to your new techniques. Well done! You also need to remember that any behavioral plans that are newly applied would need roughly a couple of months in order to see solid results. The fact that she responds positively to your new behavioral plan shows that she can adhere to instructions if there is the right incentive or circumstances or implications. This is something that ideally should be followed at school using a similar behavioral plan. However, school is altogether a different environment and there are various influences. Her friend moving away could have affected her and encouraged more chatting especially if she is her best friend. I would advise you to inquire a bit more about this behavior with the school e.g. are there any particular classes that she is more chatty or what time of the day. Is she chatting with a particular child e.g. her best friend or others as well? Is there a chance that she may find the work boring or too hard and chatting is a way to avoid it?Could she be moved to the front of the class and away from any distractions such as windows?I would not advise you to focus on the certificates or stickers from school. She has shown improvement at home and this is a big achievement on her behalf. If these difficulties at school persist and the school is not doing anything to address them, then you could possibly expand your plan at home to include positive behavior at school. e.g. include a reward for being concentrated at school, finish her tasks on time as the other children or follow the instructions given. You would need to target specific behaviors. The rewards again could be something that she enjoys and values. Give her plenty of positive attention and positive encouragement for the good work she is already doing. It will take some time to address all these little misbehaviors so be patient and engage school if you can in your efforts. let them know about your plans at home and ask them to support you with those if they can.
I hope this helps
All the best