Your partner is BU but this is a textbook case of why 50/50 co-parenting can only really work well if it's an amicable separation and there is some degree of flexibility and cooperation. If there's any need for court orders or rigid times/dates then co-parenting is not going to work, especially in times of emergency where there is no flexibility.
We have no idea what the back story is but if you're at a stage where there is a court order and rigid handover times, even in holidays, then there is clearly some issue with one or other parent somewhere, whether it's unreasonable or not. In this case it seems that your DP isn't bothered about his DC going to paid childcare, it's that he is being awkward about it because he doesn't know what his Ex is doing and she is refusing to give him a detailed itinerary of her movements, which on the surface looks unreasonable but could be perfectly justified (eg if he has a history of trying to be controlling via demanding this information and then holding it against her if she deviated from the plan, so now she just tells him "Drop the children at X place at Y time" as there's no need for discussion or negotiation)
If he'd come at it from the angle of "Hey, if you're needing childcare then I'm happy to keep the kids with me till you're back, can you just give me a ballpark idea of your itinerary so I know timings and what to tell the kids?" then YWNBU as his concern for her information is child centric - what information does he need to know in order to shape the day for the children? - as opposed to just wanting to know information for his own nosy reasons and because he disagreed with her decision to use childcare.
It might have been that she didn't want to tell him her exact itinerary because she didn't know, and she didn't want to ask him to keep the DC longer in case he tried to use it against her so arranged the childcare with a known CM so that she had the flexibility she needed - perhaps she knew she was flying in at X time but couldn't guarantee the exact time she would be home due to potential flight delays, airport delays, traffic/transport from the airport etc so rather than having to keep giving her Ex extra information and possibly changing plans she decided to keep his involvement to schedule (or drop off at 3pm) and have the flexibility with the childminder instead.
ExDH and I have a similar set up but the difference is that we split because we just weren't working as a couple but we're happy to be friends and any parenting is done jointly and amicably. DD spends every other week with each of us, she goes to school from one house Monday morning and home to the other house on Monday evening and has a full week with each parent. Holidays we will generally sit down and discuss our specific needs/wants so that we both have roughly 50/50 but we are happy to change the blocks to accommodate each other so for the six week holidays we both get 3 weeks each but in staggered blocks rather than EOW eg if ExDH wants to take DD on a 2 week holiday in the summer then I have no issue with switching my weeks to accommodate that, especially if he is happy to take those two weeks during a time I have unchangeable plans wherever possible.
In this situation - IE if one or other of us had to leave the country as an emergency then we would happily inform the other of our plans and general itinerary (eg flying out Friday, back home Sunday night but not 100% sure on exact times) and there would be a discussion as to whether the resident parent was happy/able to flex their arrangements to accommodate covering for the other parent, and only if they couldn't would the other parent make childcare arrangements.
So in this situation if ExDH was going abroad he would tell me "Hey, I have to go away this week, not 100% sure on timings but I'll be back Friday evening. Are you OK to have DD till I'm back?".
If I didn't have plans then I'd just say "No worries, let me know when you're home and I can drop DD off or if it's too late then I'll drop her off in the morning".
If I did have plans then I'd say so but offer to try and figure out childcare for him if it was easy to do so "Sorry, I have theatre tickets in <city 2hrs away> for Friday and I don't want DD on her own all day and night as Im going early to shop/have dinner and won't be back till late ... Do you want me to see if my parents/siblings can have her sleep over?" and only if that wasn't an option, or ExDH initially responded "No worries, I'll check with my family first and get back to you" and then it didn't work out, would either of us look to the paid childcare route.