All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad, sallallaahu ‘alayhi wa sallam, is His slave and Messenger.
Kneeling on one’s knees was the custom of the Arabs in case of fear, or when honoring something, or as a way of being polite with or showing humility to the one who is kneeled to.
Ibn ‘Attiyah said about kneeling: “(It is) the posing of a person who is blameful, fearful and full of veneration. A Hadeeth reads: "Then ‘Umar kneeled on his knees."” [End of quote]
Kneeling on one’s knees was mentioned in several Ahaadeeth, one of which was reported in Imaam Ahmad's Musnad from Abu 'Umar Ziyaad ibn Muslim from Abu Al-Ash‘ath As-San‘aani, who said: “Yazeed ibn Mu‘aawiyah sent us to Ibn Az-Zubayr, and when I came to Madeenah, I entered upon so and so – Ziyaad forgot his name – and said: "The people have done what they have done, so what should be done? He said: "My beloved Abul-Qaasim (i.e. the Prophet, sallallaahu ‘alayhi wa sallam) advised me: 'If you survive until you see some of this tribulation, then go to the Mount Uhud and break your sword, then sit in your house, and if someone comes inside your house, go to your bedroom, and if he enters your bedroom, kneel on your knees and tell him, Bear my sin and your sin so you will be among the companions of the Fire, and that is the recompense of wrongdoers.” [Al-Arnaa’oot declared it Hasan (good)]
Also, some Companions knelt on their knees in the presence of the Prophet, sallallaahu ‘alayhi wa sallam. Al-Bukhari reported in an account of a dispute between Abu Bakr and ‘Umar : “So the face of the Prophet, sallallaahu ‘alayhi wa sallam, became red, and Abu Bakr grew fearful and knelt on his knees and said: "O Messenger of Allaah, I swear by Allaah, I was more wrong," repeating this twice.”
Moreover, kneeling on one’s knees was the way of some of the Salaf (righteous predecessors) when sitting in the presence of scholars. It is mentioned in the Hilyat Al-Awliyaa’ that Sufyaan said: “When he went to a scholar, ‘Amr Al-Mulaa’i would kneel on his knees and say, Teach me some of what Allaah has taught you, to implement the verse in which Allaah says (what means): {…on [the condition] that you teach me from what you have been taught of sound judgment?"} [Quran 18:66]”
All this is evidence that it is permissible to kneel on one’s knees and that it is not forbidden in principle. The act of kneeling on both knees or on one knee in front of a person, there is nothing wrong with it if it is not done to imitate non-Muslims, and if it does not involve anything else that contravenes the Sharee'ah. However, the tradition of kneeling on one knee, which you mentioned in the question, is not a Muslim custom. It is most likely a custom of the idolaters, and it is not permissible for a Muslim to imitate them. The fact that it is done in the presence of the woman's Mahrams or to avoid major problems or because the woman who is proposed to will be psychologically hurt if the man does not do so does not make it justifiable. Indeed, it is correct to adhere to what is right, and a Muslim must adhere to his traditions and customs and not abandon them just to marry a woman. It is also not proper for a man to do so since Allaah made him a guardian of the woman. Allaah says (what means): {But the men have a degree over them [in responsibility and authority].} [Quran 2:228]
To conclude, it is not proper for a man to kneel before the woman he is proposing to. Similarly, a religious Muslim woman should not be pleased with such an act, let alone be hurt if the suitor does not do so for her.
Allaah Knows best.