ujm:
The deference I give to the secular court is not because I feel they should have a role. It is the consensus of the overwheming majority of rabbonim, batei din and toanim that it is necessary to be in accordance with the court to avoid the potential for other conflict. This might be made into a stronger argument because the batei din (or any other arbitration situation) have no jurisdiction in custody or visitation. And these agreements get filed in secular court. In NYS, anything in the agreement reached in BD concerning these matters is ignored. And the court reserves the right to litigate. In other states, the arrangement might be different. So it is the batei din that defer to the court, or look to rule in accordance. It should also be noted that batei din have a glaring distinction from the courts. The goal of the psak is that both parties agree to it, and that the BD not impose its psak on the two parties. Court does not do that. It rules and imposes.
You seek that BD should rule according to Halacha. Yes, in an ideal world, that might be possible. But I draw you to the Gemora (Bava Metziya 30A) לא חרבה ירושלים אלא שהעמידו דבריהם על דברי תורה. The finality of halacha requires something else besides the strict application of halacha. I would not say this alone. This is a Gemora. In reality, BD tries to “work things out”, which is about creating compromises (also required in halacha).