Concealing the Death from the Son Until After He Puts on Tefillin

If a person's father passed away in the morning, is it preferable to conceal the death from him until after he puts on tefillin, recites the Shema, and prays?

It seems that if he cannot attend to the deceased at this time, it is preferable to wait until after the prayer if possible.

In Shulchan Aruch, Yoreh De'ah, Siman 372, Paragraph 12, it is explained: "If someone has a deceased relative and he is not aware of it, it is not obligatory to inform him; even if it is his father or mother; and about this it is said: 'He who spreads rumors is a fool' (Proverbs 10:18). It is permitted to invite him to an engagement, wedding, and any joy, since he does not know. However, if he asks about him, one should not lie and say: 'He is alive,' as it is said: 'Keep far from a false matter' (Exodus 23:7)."
In the Barchi, Siman 341, Sif-Katan 15, it is stringent about the laws of an onen, as it brings there that the Bach is stringent that the laws of mourning apply to him even if he is not aware, and the law does not follow the lenient opinion regarding an onen. Some explained his words that he prohibited only on Tisha B'Av and not in all laws of an onen.
And in the simple language of the Shulchan Aruch, that no laws apply, however, if it will delay the burial of the deceased, it seems that he should be informed and prevented from prayer, tefillin, and Shema for the honor of the deceased, as the main law of an onen exempting from commandments is for the honor of the deceased.