Your dp is an arse!
So he expects to be trusted and accepted by your ex to not only care for but live with and be a step parent to your older child but he can't extend your ex the same courtesy when he's offering to do you BOTH A favour and give you some free time together?
How the HELL is it "inappropriate"? Assuming your ex is to the best of your knowledge a safe, capable and non abusive parent (which your op gives zero indication of) then there's NOTHING inappropriate about this at all!
He needs to grow up and get over himself!
I babysat my dds younger half siblings at times when dd and they were little and my ex's wife had absolutely no problem with that! Why would she?!
My ex was the child of a 2nd marriage he and all his siblings floated between the 2 homes quite regularly and he refers to his fathers first wife as "auntie", and even my exes dads dgc inc dd known her and refer to her as "auntie" too, she and her 2nd husband were even at our wedding, they were at exes full siblings weddings too
I also know at least 3 couples where when the "new" couples went on their honeymoons the exes had all the children of the 2 relationships to allow them to go as just newlywed couples. Totally fine and the kids had a ball too.
Just as I had total trust in her caring for dd.
He's being ridiculous and frankly owes both you and your ex an apology!
Doing this would be a great way of showing the kids that they're cared for and included by all the parents around them.
That's very healthy and mature and sensible - qualities your dp seems to lack!
exactly WHY was it inappropriate if he's so determined it's "factual" then? What risk does he think your ex poses to your dd?
Classic pissing contest yep! Dick measuring in the extreme I suspect!
A LOT of people assuming the new partner doesn't know the ex when there's absolutely no indication of this, indeed it sounds as if normally co-patenting is pretty amicable and they're comfortable in each other's homes so the ex may well know ops do and dd fairly well - at least as well as a "regular" "uncle" type figure
I also think it's highly unlikely the new dp is NEVER alone with the dc that just wouldn't be practical or normal in most families
He has the right to veto a childcare provider if he is uncomfortable, as does any parent.'except her ex has no right to veto so no any parent does NOT have the right to veto
Plus as several pps have pointed out it wasn't solely HIS decision to make!
people would say he was controlling and undermining and questioning whether there were other examples of his controlling behaviour. exactly! Replace ex with "friend" or even "relative" that the dp didn't know as well as op does - would get totally different responses
I maintain - your dp is an arse!