Can a Jewish couple who divorced one another at the rabbinate remarry each other, under Jewish law, if the wife, who filed for divorce, declared, falsely, that she had “another man” (on her lawyer’s advice) at the divorce hearing, to speed things up because her husband wanted to save the marriage?

Yes, if the local rabbinical court gives the couple permission to remarry after it is satisfied that the wife had only been acting on advice, to speed up her divorce, and there was no independent evidence of adultery.  In similar circumstances, in September 2017,  Ashdod Rabbinical Court (File  ‏ 1072852/1) granted such a request from a divorced couple who had cohabited after their divorce, and even had children together after the divorce.