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Let me limit my argument to the class of
cases for which, for some times, there is a comoving reference frame
that both crews can agree on. For the case of an acceleration profile
a(tau) in which the acceleration is non-zero for a period of time tau_c
and after that it is zero, at times for which all parties agree that the
acceleration of both spaceships is zero, the rest frame of either
spaceship is such a co-moving reference frame.
The
proper length of an object is determined by finding the interval
between two events that take place at the endpoints of an object *at
the same time* in the rest frame of the object.