How do I compress a string?

It depends on how levels are built in your game and what information you need to store in the string. For example, if there are three object types players can place on a 50x30 grid, then you can encode the entire grid into a string 1500 characters long, where letters A-C represent an object and a dot represents an empty cell:

A.A...BAC.....A.B..CBB.ABC.CCB....

Another totally different approach is to upload the string to an online service like Pastebin.com

When you upload a string, you get a unique URL, which can be used to download this string. So people can share and exchange URLs in Discord, instead of full strings.

Both uploading and downloading from Pastebin can be done with AJAX.

Did you try looking in the manual?

construct.net/en/make-games/manuals/construct-3/scripting/using-scripting/script-files

Adding script files

Script files can be added in the Scripts folder of the Project Bar. Existing JavaScript files (.js) can also be imported using the Import scripts option instead.

Factorio uses zlib for compression of json data, similar to what you're trying to achieve.

A blueprint string is a JSON representation of the blueprint, compressed with zlib deflate using compression level 9 and then encoded using base64 with a version byte in front of the encoded string. The version byte is currently 0 (for all Factorio versions through 1.1). So to get the JSON representation of a blueprint from a blueprint string, skip the first byte, base64 decode the string, and finally decompress using zlib inflate.

Here is a javascript implementation of zlib

github.com/imaya/zlib.js/blob/develop/README.en.md

I think you're better off using pastebin like what dop suggested or trimming the fat out of the data as rojo said, basically make your own file format. Doing so really depends on the amount and type of data specific to your game though.

I am using dictionaries.AsJSON inside of arrays, and then the arrays are stored as JSON inside another array

It's a very inefficient way to store data, like you said escaping " with backslashes is probably making 80% of the entire string. I don't think compressing the string with zlib will magically reduce its size from 30KB to 1KB. I suggest you change the way the data is stored. Use just one array or JSON. Don't nest JSONs inside of JSONs inside of JSONs.

In this game many thousands of objects are stored in a single JSON file. I'm using a string like this, which is pretty compact and easy to read:

{
 "island1": {
 "objects": [
 {"o": "Tree,animationname","x": 100,"y": 200},
 {"o": "Rock,animationname","x": 405,"y": 112}
 ]
 },
 "island2": {...}
}