>When should I go and get this approval letter?

When you're ready to put in offers.

>As long as I pass the 3:1 ratio test I’m sure I’m in the clear but I’d like to maximize my chances of getting a big loan?

Downpayment will be the limiting factor for investment real estate

You may be under estimating how much cash you need to bring to the table depending on what you are buying.

I know the amount of cash down determines the size of the loan for sure but as it stands I plan on house hacking and qualifying for FHA loan

So it's not an investment property it's going to be a residence. Planning on maybe a 4 plex?

Correct

Makes sense if you can pull it off. Interest rates are way up now though so make sure you double check all your numbers.