I do always have irrational anxiety when spending the night away from either or both of my DSs. I catastrophise like you wouldn't believe. I'm currently in hospital - my boys are perfectly safe with their competent and loving father who is on the same page as me parenting-wise, but I can't sleep. Because they might fall out of bed and break their necks. I know DS2 had to get picked up from nursery with a temp today, so I'm convinced he's going to overheat and have a seizure in the night. I was convinced DS1 was going to get hit by a car when coming to visit me today, despite being safely accompanied by DH. When DH took them away for a night to the Lake District during the school holidays I was a basket case - falling off boats into Windermere featured highly.
These thoughts are awful and really unpleasant for me and I cannot seem to stop them, although mindfulness/breathing/distraction helps a little. The problem comes when they make things unpleasant for my DH or DSs.
They had a BLAST in the Lakes with their dad.
DS1 simply adored his daddy/son trip to London and Paris in another school holiday and seeing a photo of him grinning his little toot off in front of the Moulin Rouge made the panic worth it (until they went up the Eiffel Tower and falling from a great height or terrorist attacks were going to happen).
I could not and should not rob my children of these experiences because of my own personal issues and neither should your DW. Your DD would absolutely love that trip. I work FT and don't like my DCs going away at the weekend either, but a one-off like this isn't going to break the bond forever, so that in itself isn't an excuse either. If it was every weekend, yes, but that's not what's being suggested.
Get your wife to FULLY talk about her anxieties with you - she, too, could be catastrophising, but thinks you'll mock her because she's being ridiculous. Expressing them out loud, or writing them down is definitely the best way to purge them though, rather than grinding them around in your head.
Do NOT insist she 'just enjoys herself' having time alone with your DS. It gives me the absolute red-mist rage when people do this because I know I won't, but feel obliged to force it when people suggest I must.
If you do convince your DW to let you go with your DD, BOMBARD her with photos and details of every little adorable thing she does, but just as statements of fact/running commentary. Avoid at all costs any hint of 'see? I told you she'd be fine.' My DH is just brilliant at couching this, mainly I think because he just loves sharing with me what our DCs do, and this is the place where it should come from, not first and foremost as a form of reassurance.
Finally, encourage her to look into mindfulness and counselling. How she feels is not normal and not pleasant. But at four, it's probably denying your DD a lovely, memorable trip to get better acquainted with the rest of her family.
Good luck!