Reddit mentions of Koch 3250393 3 Snatch Block Double Sheave

I think a block and tackle is your best bet. It's how humans have hauled up heavy things for thousands of years.

I'm not a mover, but I've done exactly that when moving to a new place - going up the exterior and in through the balcony. It went really well, so I also used it for moving small but heavy things, like boxes of books.

I made a 3:1 block and tackle, sort of like this: https://jimmygreen.com/purchase-systems/668-handy-billy-41-purchase-system. That reduces the load on the part that you pull by 3x (so your 180 lb headboard becomes 60 lbs). We used some climbing webbing attached with a lark's head to the railing of the unit above where it entered the concrete of the balcony floor, and attached the hook of the top block to that. That allowed us to haul the weight above the top of the railing, which is important. Otherwise you can only get it up to railing, and then have to haul it over by hand, which can be awkward and difficult. You haul from the upper position / destination.

At that 3:1, one person can haul the weight up, though it's easier with two, so that you can have someone on the ground to get it started, and then they can walk up to help guide it in over the railing when lowering it from the apex. For the heaviest stuff, if you have three persons, one person can haul while the other "tails" - they stand past the person hauling and hold the working end to keep the weight from being dropped as the hauler moves their hands. Having them take a turn around the railing (like off to the side) to introduce some friction in the system will make their job easier, especially when it comes time to hold the weight in place and lower it down. That also gives the hauler a chance to take a break mid-haul, if needed.

You don't need a cleat on it like on that one I linked to. You'll need a sufficient length of rope to be able to run it as 3:1, specifically three times the height you'll be running it, plus enough for the knot on the lower block, and enough extra to haul on. For your third floor balcony, assuming 10' per floor, you're looking at probably 110' minimum.

Your 3/8" polypro rope should be strong enough; if it's long enough, all you'll need are the blocks. Here are the ones I bought: https://smile.amazon.com/gp/product/B002T44TP2, $26 per. Those can be reeved for 4:1 or even 5:1 purchase if you have enough rope (I later used it in 5:1 mode to haul out a 400-something pound outboard engine, which felt awesome).

If you don't want to buy blocks, you can get by with a trucker's hitch (more detail on wikipedia), which will give you (depending on how it is tied) 2:1 purchase - enough that three or four guys hauling together can lift the weight - or 3:1 purchase if you have enough rope - though there's a lot more friction in these systems, costing you something like 40% of the advantage of the purchase. If you decide to go down this route, just say so, folks here love the trucker's hitch and will give you all sorts of details on it. A word of caution with it though: Galactic law stipulates that the successful use of the trucker's hitch to manage a significant weight necessitates playing its theme song.

Safety: Be sure your anchor point can handle at least twice the weigh of the things you'll be hauling up. After creating your anchor, pull yourself up off the floor with it and bounce on it a little. You really don't want it to tear out, damaging the building and risking personal safety.