Lukasz Czapla, 41, is accused of repeatedly shooting and smothering Julius in November 2020 at a house in Muirhouse
A court heard Mr Czapla was willing to admit the killing.
But his lawyer Iain McSporran QC said he would lodge a special defence if the culpable homicide plea wasn’t accepted.
Jurors at the High Court heard this defence stated that at the time Julius died, Mr Czapla was of “diminished responsibility.
It added he should be convicted of culpable homicide instead of murder.
But prosecutor Alan Cameron told the judge, Lord Beckett, that the offer was “not acceptable” to the Crown and Mr Czapla would have to stand trial on a murder charge.
The offer to plead guilty to the lesser charge came during the first day of proceedings against Mr Czapla, of Muirhouse.
Jurors also heard the contents of a legal document which states the 10 charges which he faces.
Prosecutors claim that on 20 or 21 November 2020, at an address in Muirhouse Place West, Edinburgh, Mr Czapla assaulted Julius.
The court heard the child, who was born in May 2018, was repeatedly struck with a skewer or “similar instrument”.
It is also claimed that Mr Czapla repeatedly discharged an air pistol at Julius, repeatedly shot him in the head and then placed a pillow on his face and that he did “asphyxiate him”.
As a result, prosecutors said the accused murdered Julius.
The other charges faced by Mr Czapla include an allegation that on 21 November 2020, he drove whilst breaching drink-driving laws in the Edinburgh area.
Separately, on the same date, Mr Czapla consumed a “quantity of prescribed medication” and drove at excessive speeds on the A90 road in Edinburgh whilst “overtaking and undertaking” other vehicles.
Prosecutors claim that he then failed to give way on the A90/M90 slip road which forced other vehicles to “take evasive action” to avoid colliding with his car.
Mr Czapla’s car is then said to have collided with another vehicle.
He is also accused of possessing drugs, including ecstasy and LSD, as well as having an air weapon.
Mr Czapla has pleaded not guilty to all charges.
The trial continues.
