Trying to add custom data to a plugin template that uses JS data in WordPress

I’m trying to add some custom data to a template that is causing some problems to me because is different to what I’m used to it on my project. The plugin name is Peepso, and I have customized a lot of templates and archives but there are some that are quite differente and render the data like this:

<script type="text/template" class="ps-js-member-item">
    <div class="ps-member">
        <div class="ps-member__inner">
            <div class="ps-member__header">
                <a href="{{= data.profileurl }}" class="ps-avatar ps-avatar--member">
                    <img src="{{= data.avatar }}" title="{{= data.fullname }}" alt="{{= data.fullname }} avatar">
                </a>
            </div>

            <div class="ps-member__body">
                <div class="ps-member__name">
                    <a href="{{= data.profileurl }}" class="ps-members-item-title" title="{{= data.fullname }}">
                        {{= data.fullname_with_addons }}
                    </a>
                </div>
            </div>

            <div class="ps-member__actions">
                <a class="ps-member__action ps-js-invite" data-id="{{= data.id }}" href="javascript:">
                    <span data-invited="<?php echo $force_add ? __('Added', 'groupso') : __('Invited', 'groupso'); ?>"><?php echo $force_add ? __('Add to group', 'groupso') : __('Invite to Group', 'groupso'); ?></span>
                    <img src="<?php echo PeepSo::get_asset('images/ajax-loader.gif'); ?>" alt="loading" style="display:none" />
                </a>
            </div>
        </div>
    </div>
</script>

I have found the AJAX call that handles and render the user data and the PHP function that sends the JSON data, but since this is a plugin this is not direct customizable as you already know. All I need is some way to send extra data inside the JSON in order to be able to render that specific new data since I already have my function that retrieves this data and I use it in “normal” templates with all data being handled in PHP, but this one is different since there is no “backend” data inside the archive, so for example I can’t use the User ID to retrieve any data because this is handled with the JSON and AJAX call. Thanks for your help in advance!