Your query is very interesting and is about how Atharva-Veda is not mentioned in another kandas of Valmiki Ramayana and only in Bala-Kanda where Rishishringa is telling King Dasharatha about conducting Putrakameshti sacrifice.
इष्टिम् तेऽहम् करिष्यामि पुत्रीयाम् पुत्र कारणात् |
प्रोक्तैः मन्त्रैः सिद्धाम् विधानतः || VR. 1.15.2||
"I will procedurally conduct the ritual contained in the preamble of
adharva-veda with procedural hymns, called <>putra kaameSTi i.e.,
the ritual that bestows sons, for your benefit."
We get the hint of the answer from a book called The Society of the Ramayana
By Ananda W. P. Guruge - Page no.307.The reason is mentioned there.
So, coming back to the point of the answer rest of the Ramayana only mentioning names of three Vedas i.e. Rig-Yajus-Sama. According to author that is because at the period of Ramayana, Atharva-Veda was not established or was not known as Veda although the Hymns were already known at that time and were known only as collection of Hymns belonging "Atharvans (Mantra Atharvanah)" And the older name of the Samhita was "Atharvaangirasa" (अथर्वअंगीरसा).
Below is citation from the passage of the book explaining this.
The Ramayana dose not tell us the number of Vedas which were generally
studied ; a verse in the BE belonging to the late Kaccit section
refers in an enigmatic verse to Vidyas tisrah which the commentary
explains as the three vedas. And in different contexts the Rig-Veda ,
the Sama Veda And the Yajur Veda are mentioned. But the Atharva-Veda
is not referred to as Veda although the the Hymns of the Atharvans
(mantra atharvanah) are known.
This context is very useful in assessing the position of Atharva-Veda
during the period reflected by Ramayana. It mentions together with the
three Vedas along with the Angas and the hymns of the Atharvans ; the
distinction between the vedas and these Hymns indicate firstly that
the Atharva veda was not reckoned as a Veda at that time secondly that
it was known only as collection of Hymns belonging to the Atharvans -
a title which is reminiscent of the older name of this samhita namely
Atharvangirasah. On this evidence we may consider that Ramayana like
Buddhist texts knew only three vedas as such.
We may note also that a later verse in the Balakanda of the BE
attributes to Rsyasrnga the performance of rite according to the
mantras expounded for the Atharvashirsha , which according to the
Brahmanas is the name of a brick used in building the sacrificial
So, in conclusion according to this theory the Hymns of Atharva Veda were already present there but they were not known as Vedas at that time of ramayana and the name Atharvashirsha is also name of the brick used in building altar and not Atharva Veda.
Special Note - :
It's possible that the name of the veda mentioned in the Valmiki Ramayana verse is अध्वर्युवेद - Ardvayur veda i.e. Yajurveda and not Atharva Veda. Its a translation problem. So the Ramayana isn't mentioning the Atharva veda but actually Yajurveda.