The power audit is going to be critical to this and since you have such a hard budget there are a few places you can skimp out and a few places you really can't.
Things you CAN skimp on:
Panels - Try to track down used panels pulled from a home that's upgraded or a solar farm or someone else's array that has been upgraded. The cost per watt will be pretty dramatic savings, like to the tune of $0.25/watt that I paid for my used panels VS the $0.90/watt that new panels cost on average.
Charge Controller - The Victron fanboys will tell you that it's Victron or Nothing, but there are many, MANY brands out there that will turn solar panel DC into battery DC just fine. You'll just need to be willing to give up the fancier features like Bluetooth monitoring and online data logging and dog washing and disco lights.
Batteries - While LFP batteries are a great investment overall, the up-front cost is pretty hard to swallow when you've still got to get all the other things. Look into Leisure Batteries as you can get a 100Ah-ish battery for about $100USD out the door in most places. Yes, it's only good to about 50% DoD, but to put it in reference I can buy
8x 120Ah batteries for $800 for my 24v system and get 5760Wh of usable power out of them. A pre-built LFP battery of similar capacity is over $1500USD for 4800Wh, and a DIY really isn't that much less. Yes, they may only last a few years but that buys you
TIME to get LFP's in the future.
Inverter Size - If your Power Audit tells you that you need a 1344w inverter, don't spend the money on a 5000w unit. Just get what you need plus about 20%-ish.
Things you CANNOT skimp on:
Inverter Style - That cheap Modified Sine inverter is a great deal, and you can put all the money you saved towards new fridges and CPAP's when they fry out from the dirty power. Drop the bucks and get a Pure Sine inverter that is large enough to power your loads without being taxed.
Wire - Stick with real copper wire that is rated for the higher temperatures, it's going to last a lot longer under load and be much safer.
Fuses - Get the RIGHT style fuse for your system, make sure it's the right size for your needs. The tube-type "Automotive" fuses? Don't trust those. Generic knock-offs that are 1/3 the price of the name brand? Do you really want to risk it? Remember:
FUSES ARE CHEAPER THAN FIRES! Having said that, you can save some cash by only getting the fuses you need and are comfortable with, usually being anything with a draw. Battery to Inverter? FUSE! Battery to step-down? FUSE! 12v fuse block, well obviously fuses. SCC to battery? Eh. Fuses on both positive and negative lines of everything? Seriously?
There's going to be a LOT of give & take, but the nice thing about the Power Audit form is that it can be done many times with different setups. Start with getting the usage data and then throw in the wish list of what you'd
LIKE to have in there, like the Espresso Machine and heated cat blanket and MargaritaMaster 9000 and see what it says. Then start whittling down and see where the sweet spots are. Once you have that you can start trying to find the hardware to meet your budget. Somewhere in the middle they will meet up.
Yes, it's a LOT to think about, but it's SSSOOO much cheaper than being one of those "I bought all this stuff, how do I make it work?" people.