I do battery replacements on these monocoque Ebels regularly. As far as I'm aware all movement parts are totally restricted, hence why you're not being allowed to take it apart. The watch clearly took in moisture/water that has damaged the split stem, forcing it to come out as one piece, this action of it being forced out without the release lever being depressed probably means that the keyless work is damaged and it's not engaging again because of this. So it looks like the movement is damaged. I don't think you're missing a screw, other than one of the bracelet securing screws on the back of the case? It looks like it may, possibly be needing to go back to Ebel. (which will be expensive to say the least, they'll quote for full service, stem and crown, and a new dial.)