Based on the sources [Daas Torah Heter Meah Rabbonim] and after consulting with Rav Moshe Sternbuch (who is very upset with what the Kanminetskys have done) - who read through all the relevant material on the case which I provided him - the following is clear.
Aharon Friedman can remarry without a heter meah rabbonim. That is the view of the majority of poskim. The Heter Meah Rabbonim was created to protect the wife. In this case, Tamar as moredes and going to secular court as well as declaring that she was never married to Aharon – has thrown aside the protection of the Heter Meah Rabbonim.
Tamar is clearly a moredes by deserting Aharon and taking their daughter to live with her parents. Furthermore she went to secular court without permission of the Baltimore Beis Din that she signed an agreement to obey. Finally she says she has no need for a Get since she never was married to Aharon and is living in an adulterous relationship at the present time according to the consensus of the major poskim of our time.
Additionally there is no obligation for Aharon to give a Get and he is not even required to deposit a Get with Beis Din before getting married. However it would be desirable if he did deposit a Get in case Tamar ever realizes the terrible mistake she has made and returns and accepts the full authority of the Baltimore Beis Din to decide the issues. But clearly the permissibility for him to remarry is not dependent upon providing a Get – either directly or depositing one with beis din.
Besides deserting Aharon and took away their daughter without his consent and going to secular court without the approval of the Baltimore Beis ;she compounded the problem by publicly shaming Ahron. Against the accepted halacha, she tried to pressure Aharon through shaming in the media world wide, public demonstrations, letters to have him fired from his job and even paid a poster displayed in the Washington Subway system against him. Goons were paid 60k to have him beaten up. She – with the assistance of ORA and Rabbi Schachter and Rabbi Kaminetsky was made into the poster girl for Agunos. Her reason for leaving Aharon was that she and her family thought she could do better – despite the fact she viewed him as a good father and husband – though not as sociable with her family as she wished. After failure to obtain a Get for a number of years – she switched gears and with the shameful collusion and encouragement of the Kaminetskys conducted a world wide shopping expedition to find a posek who would declare her marriage null and void. Through a phony psychiatrist report – from a therapist who didn’t even meet with Aharon – she had him declared to be totally crazy and unfit for marriage. A claim she never made to beis din or in any of the pronouncements to media while she was in her Aguna stage.
Much to the shame and disgrace of the Kaminetskys, they got Rabbi Nota Greenblatt to not only state that her marriage was annulled and that she had never been married – but he also married her to another man. Despite the world wide protest by the leading poskim of our times that the heter was invalid and she was committing adultery and despite the fact that Rav Dovid Feinstein and his beis ruled specifically at the request of the Kaminetsky - that the heter was not valid – Rabbi Greenblatt has not retracted. More importantly the Kaminetskys - while finally accepting the heter they engineered was invalid – refuse to tell Tamar that she should separate from her adulterous relationship. Tamar is saying that she has no need for a Get since she was never married to Aharon. The Baltimore Beis Din has said given these facts Aharon has no obligation at present to give Tamar a Get and there is no permission to publicly shame him into doing so.