OK, after talking this over a bit, we think that the first thing Ashitaka and Jon would each try to do is converse and develop a rapport with Lafayette and see how it goes. Sneaking around would come after that, depending on whether or not they had to do it. Both of them are pretty good hunters, and while the situation eventually went south because he didn't kill her like he was ordered to, Jon snuck up on Ygritte, who was essentially a fairly seasoned guerrilla soldier.
If this infiltration group were not given explicit orders not to contact the Commander, Jon might have waited for a very private moment and then quickly sent a text message asking for the name of an obscure distant village that Lafayette is unlikely to know, or might have asked for one before going into this, just so that he has a bit of a backstory to use in conversation. He might speak to Lafayette of being the bastard of a large family and becoming a soldier to make a name for himself, if that were to come up.
It maaaaay also come up that Jon's sympathies are probably a little more divided than most people's... I can tell you more about that if you think it would make a difference, but Jon keeps a lot of stuff close to the chest, especially when he thinks it's important to, so it might not even come up and I think it would be situationally dependent. However, if he comes into any contact at all with the royal children and interacts with them, he's kind to them (which may hint at divided sympathies); it is also feasible that he's unable to interact with them at all even if they do come into contact. tl;dr he wouldn't be out for the blood of the royal family at the time being and with things as they stand, and Lafayette might pick up on that.
Later, once Dany tells them about Marie's secret and how she alludes to "getting imprisoned last time" he would have contacted the Commander to ask what got them imprisoned there, or possibly asked among the other guards if he could -- a lot of that would depend on how they were reacting to him and how generally it seems to be known why the royal family is imprisoned there. It may also be knowledge he picked up soon after arriving in Paris or something. (I'm assuming it was probably the flight to Varennes, but he wouldn't know that.)
Jon would disguise any interactions with Dany as a flirtation between a young guard and a maid, and would not have much trouble being convincing doing so.
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If this infiltration group were not given explicit orders not to contact the Commander, Jon might have waited for a very private moment and then quickly sent a text message asking for the name of an obscure distant village that Lafayette is unlikely to know, or might have asked for one before going into this, just so that he has a bit of a backstory to use in conversation. He might speak to Lafayette of being the bastard of a large family and becoming a soldier to make a name for himself, if that were to come up.
It maaaaay also come up that Jon's sympathies are probably a little more divided than most people's... I can tell you more about that if you think it would make a difference, but Jon keeps a lot of stuff close to the chest, especially when he thinks it's important to, so it might not even come up and I think it would be situationally dependent. However, if he comes into any contact at all with the royal children and interacts with them, he's kind to them (which may hint at divided sympathies); it is also feasible that he's unable to interact with them at all even if they do come into contact. tl;dr he wouldn't be out for the blood of the royal family at the time being and with things as they stand, and Lafayette might pick up on that.
Later, once Dany tells them about Marie's secret and how she alludes to "getting imprisoned last time" he would have contacted the Commander to ask what got them imprisoned there, or possibly asked among the other guards if he could -- a lot of that would depend on how they were reacting to him and how generally it seems to be known why the royal family is imprisoned there. It may also be knowledge he picked up soon after arriving in Paris or something. (I'm assuming it was probably the flight to Varennes, but he wouldn't know that.)
Jon would disguise any interactions with Dany as a flirtation between a young guard and a maid, and would not have much trouble being convincing doing so.