There are two parts to this: would wearing a stranger's face like that of the Frey serving girl be the smart, rational thing to do here, and if yes, would Arya do the smart, rational thing here?
Would it be smart?
Possibly not based on what Arya knows. She was accosted at the gate for seemingly being a commoner the guards didn't recognise. The guards may have been pretty useless, but this does demonstrate that the guards and others don't take kindly to unfamiliar strangers walking around.
As Arya Stark, she can go anywhere. As a random commoner no-one recognises, she risks causing commotion, possibly even being mistaken for a hostile spy or a thief.
Unless of course she murders a random Winterfell resident to take a recognisable face. Book Arya is practically a textbook case study in clinical sociopathy and might well do this as casually as she killed that seemingly Stark-loyal House Bolton guard in Harrenhall; but for TV-show Arya, it'd be out of character for her to murder an innocent Stark servant so soon after coming home.
If it was smart, would she do it?
Arya is intelligent and capable of figuring out what the smart thing to do is - but she's also hot-headed, impulsive, and arrogant about her own abilities. She often doesn't choose the smart thing. Here, it'd require two things:
- Arya to plan ahead and take safety precautions: "Littlefinger is up to something. I should go on a reconnaissance mission and find out what he's doing. What situations might I encounter on such a mission? Hmm, I might need to do something like break into his room, and he might spot me. I should put a face on, as a precaution"
- Arya to recognise limits to her abilities: "...because this could be a trap or a trick, and I might make mistakes or not be as stealthy as I think I am, and it's possible I might be observed and not realise it"
These don't sound much like Arya. She sometimes makes plans, but has she ever planned for the consequences of her actions, or recognised the limits of her abilities? She's more often impulsively charging in head first, often nearly getting killed. Even when she's following her training and executing a complex assassination plan, she's easily distracted (e.g. Meryn Trant) or unable to resist risky gloating before killing her victims.
The way Arya is played is interesting from a storytelling point of view. She's presented like the type of relatable protagonist we can't help rooting for - and usually in fiction, these characters are the type whose actions tend to be what the audience imagines they might do. They tend to be somewhat generic hero characters, who do the Right Thing unless it's "for love" or some well-signposted personal weakness they overcome bravely and heroically in the final showdown.
That's not Arya.
Almost every season, the storytelling leads us down the path of rooting for Arya, thinking "Yes! She's going to kick ass and kill the baddies and look awesome while doing it!", then with a jolt she does something to remind us that she's still an impulsive, reckless child with a massive stubborn streak and an extremely black and white world view.
Season 6 was a bit of an exception - but the show loves to build up our hopes then remind us with a crash that character development isn't flicking a switch. Like with Sansa, Jaime, Dany, Jon, Stannis and many others, it shows a new side emerging, then shows us with a bump that the old personality traits and habits didn't just magically disappear overnight, Hollywood-style.
She's more likely to think: "Littlefinger is up to something. I'll follow him, right now. I'm a super smart super assassin (right?), much cleverer than he is (right?) and everyone loves me here (right?). What could possibly go wrong?"
After the Season 7 finale:
That trend of building her up to be the cool vengeful assassin only for her to go off in a wildly different direction is nicely inverted, but we still have the question, was Arya fooled by or fooling Littlefinger at this point?
There's no firm answer either way, but there's a clue in a deleted scene described in this answer: Sansa came to Bran and begged him for help before the trial scene.
The fact she needed to ask Bran for help, rather than, say, simply having another secret planning meeting with Arya, suggests that the mistrust and tension between the Stark sisters was genuine and Sansa was afraid of both Arya and Littlefinger until, with Bran's help, the sisters came together.
This would suggest Arya didn't know she was being played by Littlefinger at the moment she found this letter, and so the arguments with Sansa weren't just to let Littlefinger think he was winning: she really was angry. This also fits Arya's conciliatory comment after the trial about how she couldn't have endured the things Sansa had. She's taking back her previous barbed comments accusing Sansa of being weak: if these had just been a ruse because they thought Littlefinger was watching, this rare almost-apology wouldn't be necessary.
