According to The simple Passukim, (not medrashim) I don't see why not. He was killing all males in order not to have them multiply, if his daughter wants to keep one male, It wouldn't hurt his directive.
הָ֥בָה נִֽתְחַכְּמָ֖ה ל֑וֹ פֶּן־יִרְבֶּ֗ה וְהָיָ֞ה כִּֽי־תִקְרֶ֤אנָה מִלְחָמָה֙ וְנוֹסַ֤ף גַּם־הוּא֙ עַל־שֹׂ֣נְאֵ֔ינוּ וְנִלְחַם־בָּ֖נוּ וְעָלָ֥ה מִן־הָאָֽרֶץ׃
Come, let us deal wisely with them, lest they multiply, and it happen that when any war breaks out, they also join themselves to our enemies, and fight against us, and escape out of the land.
In fact Batya seeked out a Jewish Nurse to feed Moshe for all those months because the baby was Jewish. (many times a day for 18-24 months as this is the min. the torah recommends nursing - Ketubot 60b)
How could she do all of this with the expense of the King without his knowledge.
וַתֹּ֧אמֶר לָ֣הּ בַּת־פַּרְעֹ֗ה הֵילִ֜יכִי אֶת־הַיֶּ֤לֶד הַזֶּה֙ וְהֵינִקִ֣הוּ לִ֔י וַאֲנִ֖י אֶתֵּ֣ן אֶת־שְׂכָרֵ֑ךְ וַתִּקַּ֧ח הָאִשָּׁ֛ה הַיֶּ֖לֶד וַתְּנִיקֵֽהוּ׃
Pharaoh’s daughter said to her, “Take this child away, and nurse him for me, and I will give you your wages.” The woman took the child, and nursed it.
You can say that she gave over Moshe for that time period and only took him after. But, most Mefarshim say ויהי לה לבן is that Moshe just became an official son.
(Ibn Ezra, Bechor Shor, Rav Avraham ben Harambam).
(maybe only then was Moshe recognized by the kingdom)
וַיִּגְדַּ֣ל הַיֶּ֗לֶד וַתְּבִאֵ֙הוּ֙ לְבַת־פַּרְעֹ֔ה וַֽיְהִי־לָ֖הּ לְבֵ֑ן וַתִּקְרָ֤א שְׁמוֹ֙ מֹשֶׁ֔ה וַתֹּ֕אמֶר כִּ֥י מִן־הַמַּ֖יִם מְשִׁיתִֽהוּ׃
The child grew, and she brought him to Pharaoh’s daughter, and he became her son. She named him Moses, and said, “Because I drew him out of the water
Batya would still have to create a story to the King of who he was and how she got Moshe.
Another point, all her servants were aware of his identity. (practically she would have to kill them all)
וַתִּפְתַּח֙ וַתִּרְאֵ֣הוּ אֶת־הַיֶּ֔לֶד וְהִנֵּה־נַ֖עַר בֹּכֶ֑ה וַתַּחְמֹ֣ל עָלָ֔יו וַתֹּ֕אמֶר מִיַּלְדֵ֥י הָֽעִבְרִ֖ים זֶֽה׃
She opened it, and saw the child, and behold, the baby cried. She had compassion on him, and said, “This is one of the Hebrews’ children
The question is, did the Kingdom know. But this would be easy for a King to conceal.
This also explains why Pharaoh had Moshe ordered to be killed.
If it was his Regular grandson, at least in his eyes. What would be so wrong if Pharaoh grandson killed someone in his kingdom out of compassion.