I think he should have let him go with them as he was an extra pair of hands to help and guide them through the dangerous paths and obstacles that awaited them. Maybe he could have stopped Ron from leaving and from Harrys wand getting snapped by Nagini's unexpected visit.

I was super surprised at Lupin running away from it. I always thought of Lupin as being very grown-up and think-before-you-do person, but I guess he was scared out of his wits at this situation.
I think Harry's reaction was very typical of Harry, since when is he known to keep a cool head? It's something I would bill him to do in the circumstances. Harry's MO is to be carried away by his hot head and then feel bad about things.

I thought Lupin's reaction to Harry was uncharacteristic of Lupin, who's always a very collected person, but I guess Harry's coward comment was too close for comfort, since I am pretty sure he himself could see that he was not doing the bravest thing at that time. The fact that he does rethink and go back to Tonks shows that. I have a feeling he would always have ultimately done that (gone back to Tonks) but Harry's jibe about cowardice expedited the process.

I see the frustration coming from Harry. The things he has been struggling with his entire life, his parents not being there for him and all, and Lupin comes to him and tells Harry that he doesn't want to be a part of his son's life. Lupin has a choice to stay with his son and wife, whilst Harry's parents died for him. Harry snapped because he has lived without the love from his parents his entire life, and it triggers him that Lupin takes the wrong choice.

On the other hand, I see Lupin's frustration as well. He is concerned for his family and doesn't know what his son will become. Still, the seed has already been sown, and he made the wrong choice by leaving Tonks when she was pregnant.

I like their relationship. After Sirius' death, he was Harry's only 'family' I can say that... I was sad when he died.