One thing Makis and his faction have not taking into account is the king, and what he'd say about it.
From this chapter, it can be surmised that the king is at least amicable towards the MC, and so I can see little reason for him to believe that she'd be able to or willing to steal the tiara for such a reason. I don't think they actually have a strong enough case to accuse the MC of the theft that isn't explainable by falsifying the evidence and hearsay.
Most of the nobles will jump on it and make it difficult for the king to manoeuvre around the 'witch hunt'. Makis will get his wish and have his engagement with the MC broken, but I feel like the woman he's with is hiding something too, which will be found and ruin everything.
The king is more than "at least amicable" toward her - she's the spitting image of the late First Queen, whom he (apparently) still loves dearly.
Makis cites that lingering attachment as part of his hatred of MC; because he's convinced that his father ignores both him and his mother (the Second Queen) in favor of MC.
On top of that - MC's family is heavily involved with and knowledgeable in the gemstone business and trade - which represents the lifeblood of this kingdom's economy due to their value for the surrounding nations.
But because the king (allegedly) refuses to hand over the gemstone trade to noble families--even those knowledgeable in gemstone valuation and trade--and desires it be kept entirely within the purview of the royal family, MC was engaged to Makis to "incorporate her family into the Crown" and thus take advantage of her knowledge to prevent the kingdom from falling into economic ruin.
Which is how we get to this point - and Makis resents MC because of her mannerisms and seeming lack of emotion. He effectively feels inferior to her, and his pride has taken a hit and he's being petulant about being abandoned and ignored (in his eyes).
So he wants to break the engagement, but can't, because his father the king is so invested in getting MC into the royal family (for those personal reasons relating to the Late First Queen, and for her family's gemstone business and knowledge).
And yes - the "mistress" undoubtedly has something to do with all of this, though it's currently unclear whether it's simple ambition on her part, or some larger/more complex scheme. But she was the one who hinted at "having more fun" involving the engagement ceremony, and the implication is
she hatched this "fake national treasure plan", or at least inspired it in Makis.
But also - the king wouldn't be able to unilaterally tell off the whole nation from persecuting MC. There's an entire faction of nobles alone who despise MC because of her family's acumen and the resulting "proximity & access & favor" her family gets to/with the Crown.
At best
now he can prevent her execution, but even that's not really his doing, so much as it is this foreign prince who's sweeping her away. But the king is now
also risking access to her family business - but she solved that by asking him for their protection, in exchange for using their expertise.
Yes, there's a chance that the
hard evidence wouldn't be enough to confirm guilt in a court of law, but MC is reviled by the kingdom because of all the rumors spread about her by the prince and his faction as the "gem eating villainess". So there's a chance she or her family could be attacked at any point, or any number of "accidents" if nothing else. (In the first chapter she comments about how even the royal servants are interfering with her own personal maid, and that she could potentially see her meals poisoned as a result by those who believe all the rumors/have it out for her.)