'The love of God' is an ambiguous expression and can mean a person loving God or it can mean the love which God has towards a person.
'God's love' is what is called the Saxon possessive and means the love which God has for a person.
Context usually makes clear which meaning is intended and translations should ensure that there is no ambiguity.
But whoso hath this world's good, and seeth his brother have need, and shutteth up his bowels of compassion from him, how dwelleth the love of God in him? [I John 3:17, KJV.]
In I John 3:17 I would say that both meanings can be seen.
We know that love is of God. There is no other source of real love.
Beloved, let us love one another: for love is of God; and every one that loveth is born of God, and knoweth God. [I John 4:7, KJV.]
And hope maketh not ashamed; because the love of God is shed abroad in our hearts by the Holy Ghost which is given unto us. [Romans 5:5, KJV.]
Love, from God - shed abroad in the heart by the Holy Spirit - is that love which is expressed from one to another.
It comes from God, is shed abroad in the heart by the Holy Spirit, and is expressed through one person to another person.
We love him, because he first loved us. [I John 4:19, KJV.]
So in I John 3:17, the love that is of God is also love that is expressed to God. And is the love that is expressed to others, also.
The meaning is, I would say, a dual meaning.