Scott,
Just wrap the elements that you do not want to see in a new div with a style of display none.
...
</div>
<div style="display:none;">
<div style="font-weight:bold; margin-bottom: 5px">
<span>${nls.outputSR}</span>
<a href="https://developers.arcgis.com/javascript/jshelp/ref_coordsystems.html" target="blank">WKID</a>
<span>:</span>
</div>
<div style="margin-bottom: 5px">
<div>
<input type="text" data-dojo-type="dijit/form/ValidationTextBox" data-dojo-attach-event="onChange:_onOutputSRChange"
data-dojo-attach-point="wkidInput" data-dojo-props='style:{width:"100%"}'
data-dojo-props="required: false"/>
</div>
<div class="wkid-label jimu-ellipsis" data-dojo-attach-point="wkidLabel">
</div>
</div>
</div>
<div style="font-weight:bold;" data-dojo-attach-point="labelsTitleNode">
${nls.labels}:
</div>
...
Lines 3 and 18 above were added.