The Putresti Yajna was the one which led to the birth of Bhagavan. It is so because chronologically the mention of the Ashwamedha comes first and ends with Dasharatha feeling happy and chronologically and in very clear words next comes the putresti Yajna for the purposes of begetting sons.
The description of the Ashwamedha Yajna ends with verse 55 of the 14th Sarga, the result of which is described:
Sacrifice destroys sins. It leads to heaven. It is incapable of being done by other monarchs. King Dasaratha was very much pleased after performing this great sacrifice.
-Valmiki Ramayana 1.14.55
That’s it!
It is after the above mention of the ending Ashwamedha Yajna, comes the part where Dasharatha requests Rishyashringa to facilitate the continuity of his lineage:
ततोऽब्रवीदृश्यशृङ्गं राजा दशरथस्तदा।
कुलस्य वर्धनं त्वं तु कर्तुमर्हसि सुव्रत॥
King Dasaratha then addressing Rsyasringa said "O Adherent of vows you can help the continuity of my race"
-VR 1.14.56
In response to this, the rishi assures him of begetting 4 sons and suggests the performance of the Putresti Yajna in accordance with the Atharvaveda (as stated in the question). The rishi then proceeds to pour oblations for the putresti Yajna:
तत: प्राक्रमदिष्टिं तां पुत्रीयां पुत्रकारणात्।
जुहाव चाग्नौ तेजस्वी मन्त्रदृष्टेन कर्मणा॥
To help Dasaratha beget sons, brilliant Rsyasringa commenced putriyeshti, a sacrifice for begetting children, by pouring oblations into sacrificial fire and chanting mantras in accordance with traditions.
-VR 1.15.3
As per the next verse it is at this very sacrifice that the devatas assemble to take their share, have a whole discussion, following which from this very sacrifice a prajapatya Purusha comes taking the Havis that will produce children. While granting it to Dasharatha he says:
भार्याणामनुरूपाणामश्नीतेति प्रयच्छ वै। तासु त्वं प्राप्स्यसे पुत्रान्यदर्थं यजसे नृप॥
O King you are performing this sacrifice for the sake of sons. Give this to your worthy consorts to consume it. They will bear you sons".
-VR 1.16.20
Now we know that the sacrifice referred to by the Prajapatya to is the Putresti only as it comes in perfect chronology much after the Ashwamedha has finished and the Putresti Yajna has begun. Secondly as mentioned above पुत्रीयां पुत्रकारणात् - Putresti for the purpose of begetting progeny is the Yajna which the king is performing and is rightly referenced to by the Prajapatya Purusha as यदर्थं (पुत्रकारणात्) यजसे नृप.
The queens get pregnant with the Payasam given by this Prajapatya Purusha:
Then the excellent consorts of the king who glowed like fire and the Sun, having consumed the choicest payasam, became pregnant in a short time.
-VR 1.16.31
Thus we can clearly see that it is the Putresti Yajna which gives the fruit of Payasam to the queens, eating which they get pregnant.
Another simpler proof from the Ramacharitmanas:
सृंगी रिषिहि बसिष्ठ बोलावा। पुत्रकाम सुभ जग्य करावा॥
भगति सहित मुनि आहुति दीन्हें। प्रगटे अगिनि चरू कर लीन्हें॥
Vasishtha called Rishyashringa and got a Putrakāma Yajna conducted. The muni gave oblations with devotion and Agni emerged taking the Charu in us hands (the very Charu eating which the three queens get pregnant)
-Ramacharitmanas Balakanda Doha 189.2
Translation from: here
