You can achieve this specialized diff functionality by defining a new file format conversion rule in beyond compare. This example was conducted in the Windows OS.

Step 0: Create a python conversion script to render the formatted json. Save the following python script somewhere on your harddrive

import json
import sys

sourceFile = sys.argv[1]
targetFile = sys.argv[2]

with open(sourceFile, 'r') as file_r:
    # Load json data
    data = json.load(file_r)

    # Write formatted json data
    with open(targetFile, 'w') as file_w:
        json.dump(data, file_w, indent=4)

Step 1: Navigate in the BeyondCompare menu to: Tools-->File Formats...

Step 2: Create new file format entry by clicking on the + button and select Text Format

Step 3: Enter *.json into the file format's Mask field, and any description that will help you recall the file format's purpose.

Step 4: Define the file format's conversion settings. Select the Conversion tab and select External program (unicode filenames) from the pull down. In the Loading field write the following shell command

python C:\Source\jsonPrettyPrint.py "%s" "%t"

Step 5: Press the Save button and optionally rename the file format by right clicking it in the File Formats Name and Mask table.

Further specializations of the json dumping could be considered by looking at the python documentation, eg sort_keys=True