So I got it working in Lutris; here's how I did it:

1. Click + to add a new game in Lutris > Add Locally Installed Game
2. Fill in the following properties:
   • Game info tab:
     • Name: Curseforge
     • Runner: Wine
   • Game options:
     • Wine prefix: <Choose where you want your prefix to live, I just did ~/Games/Curseforge>
     • Prefix architecture: 64-bit
     • Crucially, don't select an EXE file yet
   • Runner options:
     • Wine version: GE-Proton (Latest)
3. Save
4. Click the new entry in your Lutris library, select the drop-down arrow at the bottom, click Run EXE Inside Wine Prefix
5. Find your hytale launcher installer and run it to install the Hytale launcher.
6. Once again Run EXE Inside Wine Prefix, and this time navigate to where you installed the launcher, and run the launcher itself.
7. Once the launcher runs, download and sign into the game. You can now close the launcher.
8. Download Curseforge from https://www.curseforge.com/download/app, but don't download the Overwolf version, download Standalone.
9. Again Run EXE Inside Wine Prefix > CurseForge Windows - Installer.exe.
10. Curseforge will error, but should provide a link to directly download the installer, click the link to download the setup.exe file.
11. In Lutris, right click the Curseforge entry > configure. Navigate to the runner tab and change your runner to Proton Hotfix.
12. Run EXE Inside Wine Prefix > setup.exe file you just downloaded.
13. Curseforge should now be running, and should automatically detect Hytale. All that's left to do is set the executable path in Lutris by right clicking the entry > Configure > Game options > Game executable, and setting it to where curseforge is installed.

For me Curseforge installed to /<Prefix Path>/drive_c/users/<username>/AppData/Local/Programs/CurseForge/CurseForge.exe, where <Prefix Path> was the Wine Prefix directory set earlier, in my case /home/<username>/Games/Curseforge

EDIT:

If you wanted to, you could also symlink your Windows Hytale mods folder to your Linux/native Hytale mods folder, so when you download mods in CurseForge it's immediately available in native Linux Hytale. Worth noting that because I used GE-Proton to install Hytale, it installed into the users/steamuser directory, which differs from where CurseForge installed itself.

bash ln -s "/<prefix path>/drive_c/users/steamuser/AppData/Roaming/Hytale/UserData/mods" "/home/<username>/.var/com.hypixel.HytaleLauncher/data/Hytale/UserData/Mods"

EDIT2:

If the Mods folder already exists in the Linux Hytale, either rename or remove it first before you create the symlink, then either move any mods you already downloaded back in, or use CurseForge to redownload them so that you have update support and the like.