I don't think Mr Martinez would have Jodi examined by a Psychologist or Domestic Violence Expert (yes, the brief Jail type interview to determine if she is suicidal would be done on intake just to see where to put her)
Does anyone know if the Prosecutor has had her seen?
The reason I don't think they did is, even though she is in their "Jail" system they have to go through her Attorney's to get anywhere near her for anything including asking her a simple question. To have had her examined would take a court order I think. AND the huge risk is they have a renowned Domestic Violence Expert (Therapist, whatever) Interview-Examine her and that person determines she is a victim of DV. Whatever they find and put in a report, that report has to go to the Defense.
That could be a disaster. So they usually work off of whatever the Defense does and counter that simply by questioning that Expert on the stand. They can and do have Experts advising them on what they see and can derive from the Video's. So they are lending their opinions and it helps in cross examining a witness.
I'm going by working off of some cases here I was in. In a typical Defendant / Plaintiff situation a Motion was made to have a person examined by a Neuropsychiatrist. In front of a Judge all kinds of details had to be worked out (nobody was getting along) Down to what the person was allowed to eat and drink before the exam.. Date was settled, Judge signed off.
Day of exam and the exam was stopped as the Attorney's all wanted to be present in the exam to make sure the Doc didn't pull anything funny.. So exam cancelled and back to the Judge, Motion from the other side that they are to be present with their person while that person is tested. Judge said no, only the person and the Doc in the room. Ok back to the exam the next week...it takes a couple hours and Report is generated. And everyone is furious at the results, so the Attorney demands the raw notes. The actual answers his person wrote with her own pencil from which the report was generated (he suspected the Expert changed many answer and made the person appear very much one way)
Back to the judge

and he brought a Motion to demand all notes etc. Judge ordered in his favor, for the Doctor to include the actual test answer. The Doctor was contacted and he said he shreds them right away (ok, now I smell a rat)
Anyway that was an eye opener and I've never seen so many motions and what you can do and when you can talk to her, and if she can have orange juice and not coffee, and if coffee then wait 3 hours before testing. LOL
OK not kidding.
These things get ugly. I am anxious for the last home stretch of this trial because as I have seen, it ain't over til its over.......
(And a side note.. in the case above, the goal on both sides the entire time was one thing. Keep this case out of the Jury's hands, a Jury is TOO unpredictable. That part I remember. They were fighting like the devil but they both wanted to get this settled out. It was a big dollar case, landmark amount at the time in California and each stood to lose or win big. Moments before they had all the people file into the room to start picking a Jury, it settled out. It settled out when something very Illegal was finally uncovered on the Expert side)