I know you only have a few days to figure this out but I think that if you are able to find a solution that provides you with comfort and peace, it will be so worth it. I don't know much about the trip you are taking but from your posts, it seems that you will be seeking healing of some kind and if you are stressed with this, it might make things difficult for you.
Do you have boarding places in your area, or better yet, in the area of the hospital that you are going to? Some boarding places are just for cats and have great facilities that have rooms that cats from the same home can share and scratching posts and windows to watch birds etc.. Your cats might miss home but they would be safe without the possibility of escaping.

I was also thinking that maybe you could find a pet sitter that you could pay. You might find some advertised up at some animal hospitals or pet stores on their bulletin boards. I know it would be scary to leave them in the care of a stranger but perhaps your animal hospital can recommend someone that they know and trust. It might cost a few hundred dollars but could be well worth it if you know that your cats are being well cared for.

Lastly, is it possible for your cats to go and live with your mom for 6 weeks? Maybe she has a room that they could have all to themselves for this period of time? Or a basement area?
Just some thoughts.
I hope things go smoothly for you.