Bootstrap datetimepicker don't work with readonly or disabled

If your are still looking for a solution take a look at the snippet. This is exactly what i suggested in the comment, you just have to call datetimepicker with

ignoreReadonly: true

That's it.

$(function () {
  $('#datetimepicker').datetimepicker({
    ignoreReadonly: true
  });
});
.container {
  margin-top: 80px;
}
<link href="https://maxcdn.bootstrapcdn.com/bootstrap/3.3.4/css/bootstrap.min.css" rel="stylesheet"/>
<link href="https://cdnjs.cloudflare.com/ajax/libs/bootstrap-datetimepicker/4.7.14/css/bootstrap-datetimepicker.min.css" rel="stylesheet"/>
<script src="https://ajax.googleapis.com/ajax/libs/jquery/2.1.1/jquery.min.js"></script>
<script src="https://maxcdn.bootstrapcdn.com/bootstrap/3.3.4/js/bootstrap.min.js"></script>
<script src="https://cdnjs.cloudflare.com/ajax/libs/moment.js/2.10.3/moment.min.js"></script>
<script src="https://cdnjs.cloudflare.com/ajax/libs/bootstrap-datetimepicker/4.7.14/js/bootstrap-datetimepicker.min.js"></script>
<div class="container">
  <div class="row">
    <div class="col-sm-12">
      <div class='col-sm-6 col-sm-offset-3'>
        <div class="form-group">
          <div class='input-group date' id='datetimepicker'>
            <input type='text' class="form-control" readonly />
            <span class="input-group-addon">
              <span class="glyphicon glyphicon-calendar"></span>
            </span>
          </div>
        </div>
      </div>
    </div>
  </div>
</div>

Use the icon to pick your date not the text field. The text field is read only.