A locally convex vector space is complete iff every Cauchy net converges: It has to be Cauchy in each seminorm; and then has to converge in each each seminorm. So I guess you better look at: Property (P) Each pseudometric is complete.
Edit: In locally convex spaces one usually has directed systems of seminorms. So consider a system of pseudo metrics where for any two there is one which is larger then either one. And then the property could be: There is subsystem of complete pseudometrics which is cofinal, so that for any there is one in the subsystem which is larger.
This is just a proposal --- I did not check it.