If you know the MFUable sectors have availability then it is doable in one transaction. This avoids any risk of availibility disappearing, and the change fee.
Being an open jaw it HAS to be over the phone. Most agents know the process very well, some need some supervisor help. It gets decidedly more tricky (for the agent) when you are outside your home country; but again, doing it in one transaction avoids that risk.
If the agent really is struggling, and I don't know the EU centres, then it may well be worth cutting your losses and phoning again later.

Where one or more MFUable sectors are not available it really depends. Some routes I take what's on offer and hope for later. Others I sit it out (with a view to going Y if CW/F reward availability doesn't turn up in time). In OPs case this may fall foul of booking country rule.