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