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A narrator is writing a letter about the last 6 months of his father's life, a narrative which includes around 5 individual scenes, some of which have extensive back and forth dialog. How should the dialog be integrated into the narrative which is otherwise someone commentating on their experiences.

All Narrators have photographic memories! Unless you tell us they don't.

Just write them as they happened. Readers are accustomed to narrators, first person or third, having effectively photographic memories of everything that happened. The narrator of most 3rd person novels is disembodied, but speaks in the past tense:

They made their way down the hall, slowing as they approached the professor's door.

They heard the professor laugh from within. "She's on the phone," Lily whispered.

But the descriptions they give over the course of hundreds of pages are obviously beyond the capacity of any human's memory, and nobody notices that. The same goes for first person narrative.

Unless you specifically give the narrator a flawed memory of what happened, or who said what, but this should only be done if it is absolutely necessary to the plot; meaning the story won't work if the narrator does have a flawless memory.

but even if that is the case, it is better to make the missing memory selective and explicit:

I realized I'd been shot. I heard Josh yell, "Gun!", and I thought too late, and that is the last thing I remember before waking up in the Sheffield ward. I don't even recall hitting the pavement, though I think this knot on my forehead proves I did.

Then your narrator can have total recall for the rest of the story.

This is just a permission slip readers give authors, how the narrator remembers everything (or knows what they know) so perfectly is not questioned and does not break reading immersion or reverie.

I would say you should treat these letters as a seperate small story.

These stories are often written in first person perspective of the one that is writing those letters. That means you can write about the inner thoughts of the writer and converse as you would in any story. But you shouldn't write about other persons thoughts and feelings except how the letterwriter interprets them.

While I agree with @Amadeus's answer, I'd suggest thinking about three different points:

1. Style of the letter.

Decide if you want to write it as a real letter or just a chapter pretending to be a letter.

From my experience writing and receiving letters, postcards and emails, those missives have very little dialogue. Instead, they have a lot of 'summarised reported speech'. Of course it's different if you're writing a missive with a literary flair, but... I don't think I've ever seen one that actually includes an account of direct speech.

You could try an introdution such as:

I know this is going to get a bit long but I think it'll be cathartic for the both of us. For me, because I'll be able to digest what happened; and for you because you won't have to feel bad about not having been here. It's not like you could help it.

Let me start with the breakfast. I came down and Jenny was sulking so I tried to lighten her mood.

"Quit being an idiot," she said. Actually, she used a harsher word, but you get the idea.

Of course, you can go for the typical missive while having him going through the memories of the dialogue and then writing the summary, which can even reveal the author of the letter is actively holding some details back.

"I hate her!" Sue had had tears streaming down her face and all I could do was lower my head in powerlessness. "How could your sister do this to me? I never want to see her again in my life! I hope she falls off the boat and drowns!"

I took a deep breath and grabbed the pen.

I met Mary yesterday and she is a bit hurt about what happened. She actually told me she doesn't want to see you right now. You really need to apologise and try to compensate her in some way.

2. How perfect / flawless you want your narrator to sound like.

Typically, a 1st person narrator sounds as flawless as an omniscient narrator. If the narrator says they bought a blue office chair, then they bought a blue office chair. If they are describing a past event, then that past event happened as described. Period.

Personally, I prefer my 1st person narrators to be a little less perfect.

When I got home with the chair, my wife praised my good taste. I knew she'd like it. She loves blue.

"It's so comfortable," she said, trying it out. "And I love the green tone."

"What are you talking about? The chair's blue."

"No, it isn't. Ok, it's green-blue, but it's definitely more green than blue."

Or maybe...

I remember perfectly well the first time I entered the house. The evening light was coming in and the dark red curtains seemed to be on fire. There was someone laying the dinner table... Jack's mum, I think. Or his aunt, maybe. The moment my eyes fell on the painting behind the table, though, that was all I could focus on. Jack's uncle told me about it - he was so knowledgeable!

Once more facing the painting, I strive to recall the details... The bay was an Irish one the artist had visited as a youth. I think it was near Dublin or... I can't recall. But the village, the village had been based off a very famous Greek port town. Santorini? Probably. I remember it made me think of saints so it probably is Santorini.

I think it works particularly well for the narrator to be apparently flawless when descriving events as they're happening, but to introduce some degree of uncertainty when memories are involved, especially if the memories are either old or emotionally straining, and again especially if the narrator is trying (and sometimes failing) to recall exact words or expressions from a dialogue, even if it happened a short while ago.

I flopped onto the bed and went over the discussion even though I'd decided to put it behind my back for the time being. I couldn't help myself! Why did Martin have to insist I was betraying him? Because I enjoyed going to the cinema with Petey? He was just a friend! Why couldn't he see it? I was not... what was the word he'd used? Psychological... No. Emotional! Emotional cheating. What did that even mean?

3. Interspeding dialogue and narration.

If your concern is mostly about having long moments of dialogue without narration, which would make it look more like a playwright of sorts, then I suggest that references to actions and feelings are strewn about.

"And what did you tell her?"

"That it was an accident, obviously!"

"Well, it was!"

"Yes, I know. It's why I said that!"

"And what did she say?"

"She was not convinced, but she was mad. She'll be more sensible when she calms down."

Could become:

"And what did you tell her?" Mary asked, popping her knuckles nervously.

"That it was an accident, obviously!"

"Well, it was!"

She turned her back on me and I shook my head. She really should have paid attention, but that was Mary! She did as she felt like and forget the consequences.

"Yes, I know," I grumbled, wondering if she'd understand I wouldn't have said it if I thought otherwise. "It's why I said that."

"And what did she say?"

Probably hadn't. She was so self-centered!

"She was not convinced, but she was mad. She'll be more sensible when she calms down."

Or not. Everyone has their limits.