OK then you'll need to trace the problem elsewhere. Are you able to try a different modem/router ? Alternatively can you take your kit to a friend's house and try it there ? (Admittedly that would involve setting up for the new situation/phone number though, which is a bit offputting).
I'm also wondering if the wall socket is a bit dodgy, that inserting the phone alone, for some reason, doesn't nudge something out, but the splitter plug loses the connection. Maybe a loose connection there.
Can only really suggest experimenting, try to 'tie down' which bit of kit has the problem.