I sense a disconnect here between feeling fragile after bereavement and declining to want to look after the cat. Pets can be an enjoyable distraction and bring a lot of comfort at difficult times. This cat is used to living in a household where people are coming and going working shifts. Ordinarily, it may not get that much attention so it is either going to be a very independent creature just popping in for food, water and shelter (so not much care required) or it may have enjoyed being with someone who could give it some regularity of routine and attention for three months.
So why were you reluctant? Is it to do with living four and half hours drive away? Were you expected to do the nine hour round trip to pick it up and then the same to return it? That’s a lot of time and expense. If daughter wanted someone to look after the cat she should have made it clear she would deliver it and collect it and provide money for its upkeep for the three months she would be away. If she had done that would you have said yes?
As a health professional, taking three months off may not have been that popular with your daughter’s line managers. She may have been very busy trying to arrange cover and trying to get her workload as up-to-date as possible before her leave. We’ve all been there, working ourselves into a frazzle before two weeks off let alone three months. She’d assumed that cat care had been sorted. Early in your post you said you had only said "maybe" but now you were calling her to say you didn't want to do it. So somethere along the line you must have given the impression you would do it or was it still up in the air? Whichever, now, it was just another thing she had to do. Sometimes there’s a tiny little straw that breaks …
A family chat forum seems the perfect place to make a family announcement. Let everyone know simultaneously once a decision had been made about a date. Does it really matter who knows first? Isn’t the most important thing that your daughter has a partner she loves, who loves her and wants to make a life with her?
Some people want to plan and pay for their own wedding. Mine is a long time ago now but I know I did. They may want a simple affair too which isn’t going to cost a fortune. Not all wedding planning is fraught. Book a venue, invite some people, buy some clothes, organise some food and drink. It doesn’t have to be complicated.
My advice? Let the young people enjoy their trip. When they return, don’t make a huge thing of all this. Three months of adventure will have elapsed. The wedding date has been announced, someone somewhere will have looked after the cat. They may or may not want some help with the wedding but it’s up to them. If they don’t then simply see it as a mark of the hardworking, independent daughter you have brought up.