The sloka that was referred to is as follows:
इमं विवस्वते योगं प्रोक्तवानहमव्ययम्।
विवस्वान् मनवे प्राह मनुरिक्ष्वाकवेऽब्रवीत्।।4.1।।
"I imparted this imperishable Yoga to Vivasvan, Vivasvan taught this to Manu, and Manu transmitted this to Iksavaku."
This sloka was the first one in the 4th Chapter. We have study the words इमम् (this) and योगम् (Yoga) carefully.
Here, Sri Krishna was saying that "He imparted this Yoga to ...". Now, the question arises, which Yoga?
Sri Krishna was not referring to the entire Bhagawad Gita, because he had just completed a discourse on Sankhya Yoga (2nd Chapter), and Karma Yoga (3rd Chapter) and about to start a discourse on "Gnana, Karma, Sanyasa Yogas".
So Sri Krishna was referring to the already completed subject matter only, but not the entire Gita.
Now let us quickly go through important Slokas in 2nd and 3rd Chapters, which Sri Krishna was referring to.
1) वासांसि जीर्णानि यथा विहाय
नवानि गृह्णाति नरोऽपराणि।
तथा शरीराणि विहाय जीर्णा
न्यन्यानि संयाति नवानि देही।।2.22।।
As after rejecting wornout clothes a man takes up other new ones, likewise after rejecting wornout bodies the embodied one unites with other new ones.
2) जातस्य हि ध्रुवो मृत्युर्ध्रुवं जन्म मृतस्य च।
तस्मादपरिहार्येऽर्थे न त्वं शोचितुमर्हसि।।2.27।।
For death of anyone born is certain, and of the dead (re-) birth is a certainly. Therefore you ought not to grieve over an inevitable fact.
3) कर्मण्येवाधिकारस्ते मा फलेषु कदाचन।
मा कर्मफलहेतुर्भूर्मा ते सङ्गोऽस्त्वकर्मणि।।2.47।।
Your right is for action alone, never for the results. Do not become the agent of the results of action. May you not have any inclination for inaction.
4) दुःखेष्वनुद्विग्नमनाः सुखेषु विगतस्पृहः।
That person is called a man of steady wisdom when his mind is unperturbed in sorrow, he is free from longing for delights, and has gone beyond attachment, fear and anger.
5) सहयज्ञाः प्रजाः सृष्ट्वा पुरोवाच प्रजापतिः।
अनेन प्रसविष्यध्वमेष वोऽस्त्विष्टकामधुक्।।3.10।।
In the days of yore, having created the beings together with the sacrifices, Prajapati said: 'By this you multiply. Let this be your yielder of coveted objects of desire.'
6) देवान्भावयतानेन ते देवा भावयन्तु वः।
परस्परं भावयन्तः श्रेयः परमवाप्स्यथ।।3.11।।
'You nourish the gods with this. Let those gods nourish you. Nourishing one another, you shall attain the supreme Good.'
7) तस्मादसक्तः सततं कार्यं कर्म समाचर।
असक्तो ह्याचरन्कर्म परमाप्नोति पूरुषः।।3.19।।
Therefore, remaining unattached, always perform the obligatory duty, for, by performing (one's) duty without attachment, a person attains the Highest.
8) यद्यदाचरति श्रेष्ठस्तत्तदेवेतरो जनः।
स यत्प्रमाणं कुरुते लोकस्तदनुवर्तते।।3.21।।
Whatever a superior person does, another person does that very thing! Whatever he upholds as authority, an ordinary person follows that.
Now, let us examine whether these tenets were known in Srimad Ramayana era, i.e., people lived in the times of Sri Rama.
1) People of the Era knew about births and re-births. This can be verified in the following sloka from Srimad Ramayana.
कीदृशं तु महापापं मया जन्मान्तरे कृतम् |
येनेदं प्राप्यते दुःखं मया घोरं सुदारुणं || ५-२५-१८
"By what this horrible and very dreadful grief is obtained by me, what kind of great sin had been done by me in another life."
2) People knew about GNANIs, who remained unperturbed either in luxuries or in struggles.
धन्याः खलु महात्मानो मुनयस्त्यक्तकिल्बिषाः || ५-२६-४९
जितात्मनो महाभागा येषां न स्तः प्रियाप्रिये |
"Great souled ones with abandoned sins, with a conquered mind, those with great fortune, sages are indeed fortunate to whom there is no pleasure and displeasure."
3) While refuting Jabali's argument on atheism, Sri Rama says as follows:
दत्तम् इष्टम् हुतम् चैव तप्तानि च तपांसि च |
वेदाः सत्य प्रतिष्ठानाः तस्मात् सत्य परो भवेत् || २-१०९-१४
"Gift sacrifice, oblation, austerities performed and the scriptural texts have the foundation in Truth. Hence, one should thoroughly surrender to truth."
कर्म भूमिम् इमाम् प्राप्य कर्तव्यम् कर्म यत् शुभम् |
अग्निर् वायुः च सोमः च कर्मणाम् फल भागिनः || २-१०९-२८
"Having reached this terrestrial globe (the realm of action), only a virtuous act is to be undertaken. The god of fire, the wind-god and the moon-god reap the fruits of their acts."
So the knowledge that was imparted by Sri Krishna to Vivasvan transferred from generation to generation till Sri Rama. That is sure. It might have lost thereafter.
I would like to point out one core issue:
Sri Krishna never thought while giving discourse to Arjuna that he was an ordinary mortal. He was merged into the BRAHMAN and he never proclaimed that he was an incarnation of Vishnu