Lawyer William Booth said he and psychiatrist Tuviah Zabow visited Mario-Cesar Deus Yela, 48, at the hospital section of Pollsmoor prison at the weekend.
"He has potential suicide tendencies. I am worried about his emotional state and his physical well-being," said Booth.
He said they would not go ahead at this stage with an application to have him assessed by a panel of psychiatrists.
Zabow would first write a report on his emotional and psychological state.
In order to do that, he needed to access the docket for information on Yela's behaviour before, during and after the twins' death.
Deplorable prison conditions
Booth said that before the deaths, Yela apparently had a motor vehicle collision and possibly blacked out.
Prosecutor Louise Friester-Sampson said they were denying access to the docket.
They would also oppose his bail application.
With eyes cast down, Yela shuffled into the dock wearing a black shirt and jeans, his very tall frame stooped over.
At times he slowly rocked back and forth in his seat.
Booth said he was not eating or drinking much. The prison conditions were also deplorable.
"There are rats, lice, cockroaches, and one shower and one toilet for 20 people. He has to sleep with dirty blankets on the floor."
He wanted his client to be kept at Victoria Hospital so he could be monitored and administered pain medication.
Magistrate Goolam Bawa denied the request. He said the Pollsmoor medical staff should perform their professional obligations and monitor Yela.
The court previously heard that Yela had rented the holiday home for three weeks to visit his 7-year-old son and twins Maximo and Octavia.
Two weeks ago, he apparently sent a message to his ex-wife, Julia Engelhorn, asking her to come over for lunch.
The court heard that when she arrived, he told her he had killed the twins.
"She got up, went into the bedroom and discovered the twins. The third child was still at school. He had a knife in his possession. He then threatened to kill her," said prosecutor Benito Adonis at the time.
She apparently jumped up, managed to escape and tried to alert a security official at the estate.
She was not in court on Wednesday.
The matter was postponed until next week for a possible bail application.