The Iris Rev 8 might be just what you’re looking for. In any case, I think the term you’re looking for is “hotswapable” since you want DIY without soldering.
Personally, I have the Ergodox EZ and I love it. It almost meets all of your requirements but I think it’s more than 40%.
I think the Keychron V6 Max might be right up your alley.
It meets all of your requirements (100% layout, back light, quiet depending on your switches).
It’s also hotswapable so very beginner friendly when you’re trying to find switches you like. Here’s a sound test of all 3 switch options for the Keychron V6. There’s not a huge difference to me but I’m also just on mobile right now. With your budget you can also choose to try other switches too. You can use the stock switches as a starting point to see what type of switch best fits your needs. For example, note the force used to actuate (activate) the switch. If it feels too light, find a switch that has a higher force. Videos can help to identify which sound nice but keep in mind that the keyboard will also impact the sound.
Lastly the V6 can be used via Bluetooth or wireless USB for a cleaner look. I didn’t see gaming as a requirement so the lower latency over wireless connection should be fine.
There are other Keychron keyboards too if you want other options. The Q6 is very similar but is entirely made of metal, thus increasing the cost to $205 for a fully assembled board. Hippyo has an informative video on building and modding a custom keyboard using the Keychron Q6 as a base model.