Does her husband have the right not to break the fast with her so he can break his fast in the mosque?

Undoubtedly kind treatment of a wife includes the husband
taking care of his wife’s needs, both spiritual and worldly, and doing his
duties towards her. One of the first duties of a husband towards his wife is
to teach her about her religion and ‘aqeedah (beliefs) as Allaah has
enjoined. Undoubtedly your husband’s telling you that what he is doing is
what Islam teaches is not correct. This is a kind of speaking about Allaah
without knowledge, because the Prophet (peace and blessings of Allaah
be upon him), even though he had his companions and was concerned about
their affairs and meeting their needs, was nevertheless at his family’s
service and took care of them. It was narrated that Aswad said: I asked
‘Aa’ishah what the Prophet (peace and blessings of Allaah be upon him)
used to do in his house. She said: “He used to be at his family’s service
and when the time for prayer came, he would go out and attend the prayer.”
Narrated by al-Bukhaari, 644.  

And Allaah says (interpretation of the meaning): 

“and live with them honourably”

[al-Nisa’ 4:19]

This indicates that kind
and honourable treatment is the basis of married life as enjoined by
Allaah. 

It is well known that breaking the fast with one’s wife, even
if that is only on some days, is a kind of treating her kindly, especially
at the beginning of marriage, when sharee’ah calls for everything that will
help to strengthen the marriage, especially if the wife feels lonely. It is
also an opportunity to teach her in a practical manner some of the Sunnahs
and etiquette of breaking the fast. 

Based on the above, we suggest that the husband should take
care of his home and family, and look after them, and not fall short in his
duties towards them. He should understand that he will be rewarded for
taking care of his family’s affairs, more than taking care of others who are
not members of his family. Hence the Prophet (peace and blessings of
Allaah be upon him) said: “Charity given to a poor person is charity, but
charity given to a relative is two things: charity and upholding the ties of
kinship.” Narrated by al-Nasaa’i, no. 2528; classed as saheeh by
al-Albaani. 

And the Prophet
(peace and blessings of Allaah be upon him) said: “Begin with those who
are your dependents.” Narrated by al-Bukhaari, 1360; Muslim, 1034. This does
not mean that he is Islamically obliged to break the fast with his wife
every day, but undoubtedly it is an act of kindness towards his wife and
family to keep them company and reduce their loneliness, and to be with they
when they need him in some of their affairs, especially since the questioner
says that she is tired because of pregnancy. It is also an act of kindness
towards one’s wife and family to be gentle and kind to them, to keep them
company and to look after them. It is not an act of kindness to do as some
men do, spending the evenings in the company of their friends and not caring
about their families or wives. We ask Allaah to set the affairs of the
Muslims straight. And Allaah knows best.