Passing a path ("key1.key2") from a bash variable to jq

I am having trouble accessing bash variable inside ‘jq’.
The snippet below shows my bash loop to check for missing keys in a Json file.

#!/bin/sh
for key in "key1" "key2.key3"; do
  echo "$key"
  if ! cat ${JSON_FILE} | jq --arg KEY "$key" -e '.[$KEY]'; then
    missingKeys+=${key}
  fi
done

JSON_FILE:

{
  "key1": "val1",
  "key2": {
    "key3": "val3"
  }
}

The script works correctly for top level keys such as "key1". But it does not work correctly (returns null) for "key2.key3".

‘jq’ on the command line does return the correct value

cat input.json | jq '.key2.key3'
"val3"

I followed answers from other posts to come to this solution. However can’t seem to figure out why it does not work for nested json keys.

jq -e ".$KEY"