You have made a good argument for Lazarus being the Beloved Disciple. I agree that the scriptural evidence points in that direction.
Martha and Mary along with their brother Lazarus lived in Bethany not far from Jerusalem. Their home village figured prominently in the Gospel of John. Luke also records Jesus visit to their home.
(Luke 10:38-42) 38 Now as they were going their way he entered into a certain village. Here a certain woman named Martha received him as guest into the house. 39 This woman also had a sister called Mary, who, however, sat down at the feet of the Lord and kept listening to his word. 40 Martha, on the other hand, was distracted with attending to many duties. So, she came near and said: “Lord, does it not matter to you that my sister has left me alone to attend to things? Tell her, therefore, to join in helping me.” 41 In answer the Lord said to her: “Martha, Martha, you are anxious and disturbed about many things. 42 A few things, though, are needed, or just one. For her part, Mary chose the good portion, and it will not be taken away from her.”
Martha obviously felt close enough to her Lord to address him as she did. This family was close to the Son of God.
(John 11:1-4) . . .Now there was a certain man sick, Laz′a·rus of Beth′a·ny, of the village of Mary and of Martha her sister. 2 It was, in fact, the Mary that greased the Lord with perfumed oil and wiped his feet dry with her hair, whose brother Laz′a·rus was sick. 3 Therefore his sisters dispatched word to him, saying: “Lord, see! the one for whom you have affection is sick.” 4 But when Jesus heard it he said: “This sickness is not with death as its object, but is for the glory of God, in order that the Son of God may be glorified through it.”
Lazarus' sisters called him "the one for whom you have affection" this is an interesting reference because Jesus loved all his disciples. But it seems Lazarus was in some way singled out as one Jesus had affection for. Perhaps Lazarus was a young man too young to be an apostle but one who drew Christ toward him. Love, sincerity and humility are traits that invite love and affection.
If he as still a young man lived with his sisters then it made perfect sense for Christ to send his mother home with Lazarus who lived with women who could comfort her.