@abconline, idk if it's what you're asking, but I had fun thinking about it anyway, so here you go.
So far MC just wants to find out where a victim's head is bc a kid asked him to, not to mention that she refuses to communicate with anyone who reaches out to her and has only been replying to that kid's letters (impersonating him). He tries to play it cool to approach her and get the info, thinking she's hella dumb, but from the get-go she indirectly challenges his claims and story, and almost walks out on him, convinced he's full of shit. He's caught off guard and makes a big show to surprise her and get her to stay by outright proposing.
So far, what we can tell is that Natsume is trying to get closer to her to get the information by any means necessary (hey! I'm just an ordinary government employee guy that is somehow catching your interest! No ulterior motives, just love!), but along the way is starting to suspect what is publicly known about the case, because what he's observed of her doesn't match anything he's been told.
We're still in dark as to who Shinju is of course, but we can at least tell that she's quite clever, quick to think on her feet, extremely perceptive, manipulative and can likely read people really, really well (like when we see it from the lawyer's pov, the only person so far who claims to really know her, and it's immediately revealed that she played him from the moment they met in court again, perhaps even from the very first time as a child, to get out of an abusive household). From how cautious and suspecting of Natsume that she's been, with the questions and investigations that are coming from her side, we know that she suspects him of having an ulterior motive and is trying to find out what tf he wants from her, if it isn't the heads as he claims (the moment she suspected him of not being the same person writing the letters, she lost interest). For now, what we don't know is why she's playing along with his charade. It's clear that there's something she wants and think she can gain from playing along with him, but we don't know what it is or how he can be used to achieve that.