Both your first and second versions are not idiomatic, at least not in US English.
He gave her a ring while proposing marriage
He gave her a ring while proposing marriage to her
are idiomatic although awkward. What is being proposed, that is suggested, is a marriage.
Your third skews a stock phrase that is archaic in form, "on bended knee." Because it is a stock phrase, altering it sounds odd.
Some of what sounds natural relates to modern conventions involving proposals of marriage in the U.S. In a usual situation, a man offers to give a ring to a woman if she agrees to marry him; the offer and acceptance are a ceremony patterned on the old Latin phrase Do ut des: I give so that you will give. The gift is not technically made until she accepts the proposal of marriage.
Thus, what are more characterizations are
He proposed
He proposed by offering her an engagement ring
He proposed marriage
He proposed marriage by offering her an engagement ring
He offered her an engagement ring to propose
He offered her an engagement ring to propose marriage
On bended knee, he proposed
On bended knee, he offered her an engagement ring