Double Spy

My interpretation is that what she did is kinda right.
The castle is torn between 2 Factions. The prince is the only child of the Emperor but his mother being a maid and not a high born lady makes it easy for him to be overlooked as the heir if the Empress gives birth to another Prince.

So if the Empress gives birth to a boy then they will be the heir but that's IF it's a boy born. So it's the long waiting game until the Empress has a boy or if the Prince is selected as the Emperors official heir.

MC is loyal to the Prince since she knows the story but realistically right now as a kid he has no power.
The staff don't hold him in high regard and making it know they are not on his side.
If she made her Alliance know then she would be put in danger. The Prince has no power to protect her with against enemies outside and inside the castle.

So best bet get in the good gracies of the on a who does hold the power at this moment, the Empress.
She is assured safety and living well in the palace. Which means better food, supplies, that she can easily share with the Prince.
Just until the prince gains his own political power.

Now for the info she told, she just told the Empress that the prince was sneaking out to trains with mercenaries. Which by all means is very vague.
Since she never said which group of mercenaries and where. Which the Empress can make her own assumptions. Since she could be thinking that being trained by mercenaries is not much help training in sword fighting. Since mercenaries are looked down on she might think he's learning useless skills.

Also by being in the Empress "side" she can protect the Prince in her own way. Gaining info for him and even steering him in the right direction to build up political power.

Well I mean thats how I see it. I guess we have to wait until the next update. ╮(￣▽￣)╭