all of the comments, including any deleted ones (deleted by the user and deleted by mods/admins)
You probably will not have access to this.
Is there any other method I can do this without resorting to manual saving?
Only if you are an Admin. https://join-lemmy.org/docs/index.html
The data seems to be stored in a postgres database, and could technically be queried.
And if I have no choice other than to save each and every post and comment manually? How should I be doing it?
It depends on your need, if it is for legal purpose, it depends on your jurisdiction. Maybe web archive is enough for this case, https://archive.org/.
Otherwise, you can use the API as others mentioned. Or you can use Selenium/Scrapy/BeautifulSoup to scrape the website.
If you would rather not program something with code, look into browser add-ons that scrape websites, they are mostly visual, and you click on the things you want to save or navigate into. I am not familiar with them to recommend you something, but there are plenty of videos on how to use them.
That said, depending on the security of your instance, your ip/account might get flagged.
Talk with the Admins of your instance first and express your intentions, maybe they can help with what you need.
I don’t think it would be taboo to save something that is public available, as you can just as simple visit those pages and print them to pdf for example.
The same goes to things you have access with your account, and it is not bound by a non-disclose agreement (you can still save, but not broadcast it - depending on the laws of your jurisdiction).
I looked into the lemm.ee default profile page, and you might have success with the tools I mentioned.
You just need to navigate these links, and make the tool open the post names.
https://lemmy.ca/u/megane_kun@lemm.ee?page=1&sort=New&view=Posts https://lemmy.ca/u/megane_kun@lemm.ee?page=1&sort=New&view=Comments
You will need to take in account navigation with “next” buttons, and some pages need to be scrolled down to load all the comments.
If you don’t want to contact the Admins, and you are not pressed for time, put a delay on the tool’s web requests to not overload the servers. The browser add-on might avoid that because it mimics a more natural way of navigating a website.