@Don Fisco, this is a complicated endeavor, but most of my clients have complicated aims, and I love a challenge.
Firstly, you want to talk to a real-estate lawyer about the in's and out's of a tenancy-in-common deed. (I think I can get a solid referral for you.) A tenancy-in-common (TIC) is when multiple parties collectively own a piece of property and have assigned use of different units within the property. TICs are becoming more common in LA these days; they're like condos, but the structure is different.
I'm not a TIC expert, but from what I understand, adjusting the ratio of ownership is difficult. I believe the different parties can start at difference percentages of ownership, but to change those percentages, you'd need to create superseding deeds... which is doable... but this is where you need to speak to a TIC-experienced lawyer about what's possible and what the costs will be.
Forming an LLC would make adjusting ownership percentages easier, but I doubt you'd be able to get a loan. When you buy a property as tenants-in-common, everybody in the group is essentially combined to qualify for the mortgage. Between the four of you, you make about $510,000/yr, which will qualify you for a sizable loan. If you form an LLC, you'll have a new company with $0 of income. Good luck getting a bank to lend $1M to an LLC with $0 of income!
So, there's a structure through which you can do this (a TIC deed), and you definitely need to consult with a real-estate attorney who has TIC experience to understand how to go about doing this.
Now let's talk about what's possible with the numbers you've shared...
You presently have $640-840K for a down payment and a combined income of $510K. With that income, you should be able to borrow upwards of $2M. Add the down payment, and you're looking at a healthy budget.
Two problems:
You'll need to buy a fourplex or smaller. Once you get to five units, the loan structure changes (it becomes a commercial loan), and you won't be able to qualify with your collective income.
A fourplex means no rental income because your group needs four units, right? There is a workaround: ADUs! Multifamily properties are allowed two detached ADUs and a third ADU, if possible, inside the existing structures. So, without getting into the weeds, you could buy a fourplex, then build 2-3 ADUs for rental income. You have quite a bit of capital for the down payment, and I'd suggest allocated a chunk of it to ADU construction.
The second problem is finding a vacant building. It will be very hard to find a vacant fourplex in LA. If you buy a building with tenants, there are ways to remove them. Namely, I'm thinking of utilizing to Ellis Act to remove the tenants. The only catch is that you can't put any of the four units back on the rental market for a period of five years. If everybody's comfortable staying put for the next five years, you're good.
This remind me of another factor you'll have to keep in mind: Ellis Act evictions require 120 days' notice. That's four months. And you'll probably need to do some renovating to any fourplex you buy; that's another chunk of time.
So, in conclusion, this plan is doable. I believe the legal structure, financing, properties, and path to living in the building all exist. However, it will be difficult! It will be challenging to find a suitable property, then very challenging to go from opening escrow to, months and month later, moving in. You up for it?
Best,
Jon