For now, I am going with this jQuery hack. This hides the cog wheel menu (because I personally don't need it) and puts the edit comment link next to reply.

This does not affect brand new comments just submitted via ajax. But, once the page is refreshed, or the next time you visit the page, the edit link changes appear. Which is better than no solution at all... for now.

I'm sure you can add to this to also target ajax events. But, I'm not sure which ajax event to target.

jQuery(document).ready(function($) {
    // Loop through each comment or reply block
    $('.wpd-comment, .wpd-reply').each(function() {
        // Find the reply button within the current comment/reply block
        var replyButton = $(this).find('.wpd-reply-button');
        
        // Find the edit button within the current comment/reply block
        var editButton = $(this).find('.wpd_editable_comment').first();

        // Check if the edit button hasn't already been moved and move it after the reply button
        if (!editButton.hasClass('edit-moved')) {
            // Move the edit button after the reply button and mark it as moved
            editButton.insertAfter(replyButton).addClass('edit-moved');
        }
    });

    // Apply styles to all moved edit buttons explicitly
    $('.wpd_editable_comment.edit-moved').css({
        'margin-left': '35px',
        'font-size': '14px',
        'color': '#999999',
        'cursor': 'pointer'
    });

    // Hide the wpd-tools within the wpdcom container
    $('#wpdcom .wpd-tools').css('display', 'none');
});

I've added mutation observers to ensure this all works as a new comment loads as well. I know there are better ways to do this (if I had more knowledge about this plugin). However, this does everything I wanted.

jQuery(document).ready(function($) {
    // Function to move and style Edit buttons
    function moveAndStyleEditButtons() {
        // Loop through each comment or reply block
        $('.wpd-comment, .wpd-reply').each(function() {
            // Find the reply button within the current comment/reply block
            var replyButton = $(this).find('.wpd-reply-button');
            
            // Find the edit button within the current comment/reply block
            var editButton = $(this).find('.wpd_editable_comment').first();

            // Check if the edit button hasn't already been moved and move it after the reply button
            if (!editButton.hasClass('edit-moved')) {
                // Move the edit button after the reply button and mark it as moved
                editButton.insertAfter(replyButton).addClass('edit-moved');
            }
        });

        // Apply styles to all moved edit buttons explicitly
        $('.wpd_editable_comment.edit-moved').css({
            'margin-left': '35px',
            'font-size': '14px',
            'color': '#999999',
            'cursor': 'pointer'
        });
    }

    // Function to hide the wpd-tools
    function hideWpdTools() {
        $('#wpdcom .wpd-tools').css('display', 'none');
    }

    // Initial load handling
    moveAndStyleEditButtons();
    hideWpdTools(); // Hide on initial load as well

    // Create a MutationObserver to monitor changes in the comments container
    const commentsContainer = document.querySelector('#wpdcom'); // Replace with the actual ID/class of your comments container
    const observer = new MutationObserver(function(mutations) {
        mutations.forEach(function(mutation) {
            // Check if new nodes are added to the comments container
            if (mutation.addedNodes.length) {
                moveAndStyleEditButtons();
                hideWpdTools(); // Hide wpd-tools for newly added comments
            }
        });
    });

    // Start observing the comments container for child node additions
    if (commentsContainer) {
        observer.observe(commentsContainer, {
            childList: true,
            subtree: true // Observe changes in all child elements as well
        });
    }
});