I know, as people don't hesitate to remind you, that "it's not recommended to type young children" and all that -- but as you said, it's for fun, so as long as you are aware that at this point you are just gathering data and that you are forming a hypothesis rather than a conclusion, then how could there be any harm in it? In fact, as someone with a weird partial-custody arrangement thingy with my nieces -- who had no previous experience with children -- evaluating how I should interact with them on a personal/customized level is far more helpful than treating them As One Treats a Child. Because how does one even do that?!
So my nieces were probably about the same age when I started picking up on signs that eventually coalesced, but it did take a couple more years of consistent evaluation noting them in different environments/circumstances/moods/etc that I felt more comfortable identifying their types; up until then, it was mostly cataloguing patterns that I organized into different 'functions'. And yeah, at that age, it's basically their dominant function -- but what you can do (even if many sites tell you not to ) is check for the secondary function that corresponds with that dominant function.
So, strong Se would be either an SFP or an STP (Fi or Ti), that kind of thing, so you'd say, do I see signs of Fi or Ti? or with someone with strong Fe, you'd say SFJ or NFJ -- Si or Ni -- and then just keep watching them and updating/tallying the evidence as you go!
This works best if you have multiple children to compare with: I'd never been around children before, so I didn't know what was 'typical' or not; I just compared the two sisters (who are a little over a year apart), and measured how strong certain traits were in one as compared to the other. So obviously that would help pinpoint things faster! Otherwise just be prepared to keep re-evaluating as things become clearer/more obvious -- and yeah, especially once she gets to school age, you'll have a whole new set of data to integrate!
So my five-year-old niece was easy to type as strong Se-user early on, since like you mentioned with yours, she just has to touch everything, and even as a five-year-old, she's still putting everything in her mouth... but yeah, definitely motivated with physical experience and interactions, and pulls everything apart and breaks things and colors on walls and climbs and jumps on everything, even when she explicitly is told not to. It wasn't until she was four probably that she overcame her shyness and made it clear she was an extravert; now she is unbelievably loud and boisterous and impulsive and always forgets the rules/proper etiquette in her excitement; she's really a hoot who tries to work up her charm and get everyone to laugh, and thinks she's hilarious even though she has no concept of how telling jokes is supposed to actually work. (To be fair, she is hilarious just from being such a ham.)
I figured she was a Fi-user, starting with a hunch on how sensitive she was and how she reacted to criticism, and honestly it was probably subjective impressions the most, but one thing that did stand out to me was how she took things so personally she would ice out whoever she'd been offended by and avoid them or try to make them feel bad by preferring to be comforted by someone else, like definitely a grudge-carrier lol. Which is some unhealthy Fi-behavior I've witnessed before; though I'm not trying to stereotype. ^^;
So, sorry for the tangent there, just that I think the Se behavior/motivation is standing out a great deal in the description of your niece, and the secondary signs regarding precision are absolutely a Ti thing (as long as we're not accounting for neurodiversity such as Asperger's or something, which would interfere). However, the 'charm' aspect I wouldn't associate with ISTPs particularly -- I would say that is traditionally more an ESP? I mean, just taken by itself! However, I'm an INTP and I know I was way more friendly and silly and outgoing and extraverted as a child, and all the ISTPs I would compare to aren't children -- so just keep an eye on that aspect, I guess!
At this age, it's safer to hold out as to whether she's more introverted or extraverted -- not that that's as big a deal, but just that both of my nieces took additional years for that to become clear... my older niece is six, and I'm only now feeling more comfortable that she's probably an ISFJ and not an ESFJ, though I'm not totally solid on it yet. (ISFJs are pretty extraverted, though, to be fair, which isn't true of ISPs.) It's just harder when you're looking for their dominant function if that dominant function is introverted, since they will externally display their strongest extraverted function -- at least in ways that it's easiest for you to catch.
And honestly, it's more important in childhood to know their preferred functions just so you can orient yourself around their needs, and obviously that's the kind of information that you can customize as you get to know them, anyway! The absolute most important thing is that you respond to their individuality and embrace that. (I love my parents, but we completely clashed in types and I'm afraid that just... didn't work. But it did make me aware of how important this is -- so thank you for working to figure out your niece to help accommodate her; that is one of the most valuable things you can do for her!)
Mine are both SFs so I know that I'm not instinctively designed to 'fit' with them, so it was really important to me to analyze what helps them specifically and how to respond to them (they're so touchy-feely, ack!!), but I think it's looking like you could have an STP niece, and your being an ISTJ, you are already in-tune with a lot of the things that make her tick (like mentioning freedom and independence for sure), so that's awesome!! She is really lucky that you are making this effort and paying attention to these parts of her and are there to support her. :)
(I suspect it would stand out to you more if she were an SFP or something, just because that might chafe more against your TJ-ness, you know?? Though it depends on how accustomed you are to being cuddly and hugging and such.)
Anyway, best of luck on your typing and aunt-ing journeys -- let me know if you have any questions or would like to discuss anything!