A £250 chess set, bought secretly, on joint finances, with your own daughter roped into keeping it from you and her siblings - what a cheeky fucker, honestly, you'd be well within your rights to be absolutely livid over this.
The bit that really stands out is his own reasoning: he admits he didn't tell you because he knew you'd say no. That's not an oversight or a busy-Christmas slip, that's a deliberate decision to bypass an agreement you both made about joint spending, because he knew the answer would be "we can't justify that for one child." Knowing you'd object and going ahead anyway isn't a grey area, that's exactly what secrecy means.
A few things worth being firm about:
Getting DD to lie to you is not okay, full stop - and it shouldn't be treated as a minor detail. This isn't just "he kept a secret," he actively asked his daughter to deceive her own mother and her siblings and made her complicit in that deception. A 25-year-old shouldn't be put in the position of keeping secrets from her mum on her dad's behalf, and normalising that "it's fine to lie to Mum if Dad asks" is a genuinely damaging precedent, for her and for how she sees honesty in this family. That's not a side issue next to the money, it might be the bigger problem of the two.
This isn't really about chess sets or flat decor. It's about him deciding unilaterally that one child deserves £350 while the other two get £100, using money that's supposed to be agreed jointly.
Him calling you dramatic is a deflection, not a defence. Being called dramatic after he's the one who lied and dragged your daughter into it would send anyone into orbit. That's not an overreaction; that's the correct amount of fury for being gaslit about your own valid anger. He doesn't get to flip the story, so his behaviour is the reasonable one and your anger is the problem.
If this gets let go, it sets the terms for next time. If the response to "I hid a purchase and got our daughter to lie for me" is that it blows over with no real consequence, the lesson he takes from it is that secrecy and pulling the kids in as cover works, as long as he's willing to weather a bit of anger afterwards. Letting it go quietly doesn't just forgive the chess set, it tells him deceit is an acceptable strategy whenever he predicts you'll object.
Joint finances means joint decisions, full stop. If he wanted to spend more on one child, the conversation should have happened before the purchase, not been avoided because he predicted your answer.
You're not being over the top for feeling this angry - sneaking around on shared money and dragging your own daughter into lying about it is genuinely low, and it deserves to be called exactly that: a cheeky fucker move, plain and simple. This needs addressing directly and firmly - the secrecy and what he asked of your daughter deserve just as much pushback as the money itself, if not more.