Yes, it is absolutely correct to say that the queen is necessary for the performance of the Ashwamedha Yajna. The main part concerning the rituals of doing parikrama of the horse etc must be performed by the queen as is evident from the Shatapatha Brahmana:
the wives walk round (the horse), and thus make amends to it for that. (Verse 4)…..(the Mahishi says 5);--seed, doubtless, means offspring and cattle: offspring and cattle she thus secures for herself (verse 5).
-Śatapatha Brahmana 13th Kanda, Adhyaya 2, eighth Brahmana
Since Bhagavan is supposed to be the very ideal for Dharma - rāmo vigrahavāndharmaḥ (VR 3.35.13), it is highly unlikely that he would break the very rules he had come to give an example of, eg. The rules entail Performance of Ashwamedha with a wife. Much rightly so, there are a few other scriptures that do discuss the performance of his Ashwamedha, and not surprisingly, Bhagavati Sitaji is very much present in them or at least returns home from Valmiki Ashrama, back to her husband. They’re listed as below:-
Here Bhagavan sends Lakshmana to bring back Sitaji, when she returns, the golden image is disregarded and she is instituted for the Yajna:
रामस्तदा यज्ञमध्ये शुशुभे सीतया सह। तारयानुगतो यद्वच्छशीव शरदुत्प्रभः॥१७॥ प्रयोगमकरोत्तत्र काले प्राप्ते मनोरमे। वैदेह्या धर्मचारिण्या सर्वपापापनोदनम्॥१८॥ सीतया सहितं रामं प्रसक्तं यज्ञकर्मणि।
निरीक्ष्य जहृषुस्तत्र कौतुकेन समन्विताः॥१९॥
In the sacrifice Rāma at that time shone with Sītā, like the Moon having rising lustre in the autumn along with the star (Rohiṇī). With his chaste wife Vaidehī, he performed the sacrifice removing all sins when the lovely time had come. (People) seeing Rāma engaged with Sītā in the sacrifice, were, being full of eagerness, very much delighted.
-Padma Purana, Pātāla-Khaṇḍa, Chapter 67.17-19
Jaiminiya Ashwamedha Parva
In this, the fight between Bhagavan and his sons occurs, seeing which Valmiki intervenes and says these are yours sons and if you consider Sita to be faultless, you should take her too. After Rama goes back to perform the Yajna:
यज्ञोत्सवे वर्तमाने वाल्मीकिर्मुनिपुङ्गवः। सीतां नीत्वा पुत्रयुतां संस्थाप्य रघुसन्निधौ॥८१॥ रामः पुत्रयुतो जातः सीतया सहितः स्थितः। मुनीन् विसर्जयामास यज्ञान्ते च पुरस्कृतान्॥८२॥
While the Yajnotsava was going on, Valmiki the best among sages brought Sita along with her sons and established her in the company of the scion of Raghu. Rama alongwith his sons and wife bid farewell to the sages at the end of the sacrifice keeping them ahead.
-Jaiminiya Ashwamedha Parva 36.81-82
The Srimad Bhagavatam
What’s more, the Srimad Bhagavatam itself talks of Sitaji present while Rama is performing his Yajna:
Śukadeva Gosvāmī said: Thereafter, the Supreme Personality of Godhead, Lord Rāmacandra, accepted an ācārya and performed sacrifices [yajñas] with opulent paraphernalia. Thus He Himself worshiped Himself, for He is the Supreme Lord of all demigods. (Verse 1) … After thus giving everything in charity to the brāhmaṇas, Lord Rāmacandra retained only His personal garments and ornaments, and similarly the Queen, mother Sītā, was left with only her nose ring, and nothing else. (Verse 4)
-Śrimad Bhagavatam 9th canto, Chapter 11
We know that this is an Ashwamedha Yajna because the dakshina given by him in verse 2 and 3 (I.e. the directions to each priest) is the same as what has been ordained (स्वयंभूविहिते पुरा) for an Ashwamedha Yajna as per Valmiki Ramayana 1.14.41-42
The Ramayana itself talks of Sitaji present while Bhagavan is departing for his abode at the end of his leela. If she had been abandoned and wasn’t present for his Yajnas (as a wife should), how was she accompanying him back home?:
रामस्य दक्षिणे पार्श्वे सपद्मा श्रीरपाश्रिता। सव्ये तु ह्रीर्महादेवी व्यवसायस्तथाग्रतः॥६॥
On his right, walked Shri with her lotus, on his left, was the great Goddess Vyavasaya [i.e., The Goddess of the Earth]
-VR Uttarakanda 109.6
So well, when so many texts talk of the fact that Sitaji was present during the Ashwamedha Yajna performed by Bhagavan, surely her abandonment is a mere fairytale.
To therefore answer the question, he did not perform the Ashwamedha without her.
Note: One should focus on one more point where Rama never called back Lakshmana after having abandoned him. But that wasn’t so in the case of Sita. If he did abandon her he wouldn’t call her back, it’s not a game.