I'm not sure what Gastroschisis is but I'll give you my experience of having a c-sec, I had to have an emergency one after 20 hours of labour. Afterwards DH could hold DD2 but I couldnt obviously the epi is quite a lot stronger than the standard and I wasnt capable not to mention it makes you (or made me) shake quite a lot.
Then I was sent off to recovery where about 2 hours (from blurry memory) I was able to hold her.
Having said that Im not sure if we were allowed a camera in theatre i know we werent allowed the video camera not sure about still but given your circumstances im sure whilst they are stitching you up (IF you have the c-sec) you will be able to get a few snaps as a family.

I wish you all the best, although ive had a c-sec Im still scared about having another as Ive been told I'll have to with the twins