Anything with a variable speed compressor and two tubes to the window. The variable speed is quieter. The two tubes mean it doesn't steal cool air from inside and exhaust it outside.

I still think they are all loud trash, but that's the best you'll get for your situation.

Doesn't really apply to your setup, but I just got a u-shaped window AC from midea and it works well with home assistant. I didn't expect it, but it also has an outside temp sensor I can read. I've been graphing outside temp all summer. Purple is the AC, blue is openweathermap. Ignore humidity, that is for a tropical/moss tank.

They have an indoor unit with the same wifi capability, linky so that one might work.

EDIT: you're right, Midea compatibility isn't standard in home assistant, nor advertised , but I added it with a custom integration. It's been working great this summer.

I'm currently using a 15 year old window unit that's just plugged into a smart plug. It doesn't sound like 100% what you're looking for, but it is an alternative

If the unit you buy has an IR remote, you could look into Sensibo as an add-on. I just installed a Mr. Cool mini split, and set up a Sensibo Air at the same time. The Sensibo integration is two parts - Homekit and API from their cloud service. Most of my automations use only the Homekit integration. Sensibo's website will also let you check to see if the AC unit you're thinking of works with Sensibo.

I'm finding it really difficult to tell whether a particular air conditioner is supported by Home Assistant

Home Assistant keeps a long list of officially supported products.

I'm sure that with enough work anything will work

Not necessarily. Lots of them will go out of their way to break any external functionality so you have to use their shitty app and they can charge you a monthly fee, push ads to sell you more products, and/or mine your data for profit.

I'm not sure about compatibility with HA (saw this via /all), but I bought the Whynter ARC-14S and a Broadlink IR universal remote + sensor cable so that I could set up some automations and/or control the AC via my phone. I think any AC with an IR remote should work in that setup.

If you're in an area that gets lots of poor air quality days while you'll be running it (my area usually gets a few weeks of terrible air in August & September due to wildfires), a dual hose AC is definitely worthwhile.

I've got a different smart GE air conditioner, but it works with home assistant, through smarthq. They have a portable one (I dont know how well it does, just that it's 'smart') https://geappliances.ca/en_CA/Air-Conditioners/GE-14%2C000-BTU-Smart-Portable-Air-Conditioner-White---APWA14YBMW/p/APWA14YBMW

I'm fortunate enough to be a homeowner, but I rented places for most of my adult live. My current home doesn't have central AC, and none of my rentals did either.

Everywhere I've lived, the mounting hardware that comes with portable ACs just didn't work for me. What I found that did work is to throw away the existing window mount and build your own. I'll take two pieces of plywood, cut them to the space that exists in my window (at one point this was a sliding door), and sandwich a sheet of insulation foam in the middle. Then drill holes for your AC tubes and screw on the mounts that came with your AC.

Also, if at all possible, avoid the single-hose portable units. You're wasting cold air. A dual-hose unit uses outdoor air to cool the unit itself, and the hot exhaust gets expelled through the second tube. Do make sure you have a screen on the inlet, though, unless you want to be cleaning out bugs from inside your AC. I have used window units, single-hose portable, and dual-hose portable units. At least in my experience, the window units work best, the dual-hose units are a close second, and the single-hose units are crap.

As far as HA integration, I recently went through the process of finding a new AC that works with HA. What I found is that everything available either requires internet access and works with proprietary "cloud" access or just doesn't have any sort of remote connection. Some of the "cloud" solutions have decent integration with HA, but I have a hard-and-fast rule that none of my IoT devices access anything outside my home network. What worked for me is buying a "dumb" portable AC with a remote control and using a Wifi-connected universal remote to provide access to HA.