Wirelessly posted (BlackBerry: BlackBerry8530/5.0.0.1030 Profile/MIDP-2.1 Configuration/CLDC-1.1 VendorID/417)
"There's no problem if both of us apply for this card simultaneously, is there?"
No, not as long as both of you are credit worthy. But if your SO is going to apply for the CSP card, I would do that first (and perhaps meet the minimum spend on that card) before applying for the Chase/Hyatt card.
If you already have the CSP card and you have met the minimum spend on that, you can go ahead and apply for the Chase/Hyatt card immediately.
Just make sure that your ICN trip is not going to be put off: the two free Hyatt nights expire twelve months after they are earned.