A male doctor is going to supervise her birth in the hospital; should she have a caesarean so that she can have a female doctor?

Firstly: 

The basic principle is that
a woman should be treated by a woman, unless there is a need to be treated
by a man. That is in accordance with the guidelines and stipulations which
we explained in the answer to question no.
120224 and
5693. Among the things we said there
is:

Priority should be given to
the treatment of men by men and women by women. When a sick women needs to
be uncovered (for medical treatment), preference should be given to a
qualified female Muslim doctor; if such is not available, the order of
preference is then a female non-Muslim doctor, a male Muslim doctor, and
lastly, a male non-Muslim doctor. If it is sufficient to be treated by a
female general practitioner (GP), she should not go to a male doctor even if
he is a specialist. If a specialist is needed, she should go to a female
specialist, but if one is not available, then the female patient may uncover
in front of a male specialist. If the female specialist is not qualified to
treat the problem and the situation calls for the involvement of a
highly-skilled, qualified male specialist, then this is permissible. If
there is a male specialist who is more highly-skilled and more experienced
than the female doctor, the female patient should still not go to him unless
the situation requires this extra level of experience and skill. 

Secondly: 

Caesarean birth is not free
of risks and harmful effects, as has been explained in the answer to
question no. 92831. Hence we advise
you not to resort to that; rather delegate your affairs to Allaah and ask
Allaah to make it easy for you to have a skilled female doctor so that you
will have no need for a male doctor. 

If there is no female
doctor and you are forced to accept a male doctor, then we hope that there
will be no blame on you in sha Allaah. 

We ask Allaah to make
things easy for you and decree good for you wherever it is. 

And Allaah knows best.