It looks like we're talking pass each other. Let's start with the basic common ground. How do you define synchronized clocks?
First, they must tick at the same rate as each other.
Second, some process will have been used to set them to a known start time. (Though this can be relaxed as long as their time as at an agreed event is known; their offsets can be accounted for.)
Where we differ is your claim that a special observer (i.e. one between the two clocks, for whom the clocks tick at the same rate) can claim they are synchronised. But no, to claim the clocks ARE synchronised - it's the clocks themselves (or observers at rest with them) that need to agree they tick at the same rate. That means they must be at rest with respect to each other, i.e. not like my Alice and Bob, regardless of what Carol says.