При установке флажка добавление #mosaic HTML в #assetSelectedList, но вы хотите пропустить класс .asset-paid-li для добавления.
$("input:checkbox").click(function(){
if (this.checked) {
$("ul#assetSelectedList").append($("ul#mosaic").html());
$('ul#assetSelectedList input:checkbox').prop('checked', this.checked);
}
else {}
});<script src = "https://ajax.googleapis.com/ajax/libs/jquery/2.1.1/jquery.min.js"></script>
<input type = "checkbox" name = "thing" value = "valuable" id = "thing1">
<ul id = "mosaic" class = "clearfix list">
<li class = "clearfix asset-free-li">Asset Free</li>
<li class = "clearfix asset-free-li">Asset Free</li>
<li class = "clearfix asset-paid-li">ASSET PAID</li>
<li class = "clearfix asset-free-li">Asset Free</li>
<li class = "clearfix asset-free-li">Asset Free</li>
<li class = "clearfix asset-paid-li">ASSET PAID</li>
<li class = "clearfix asset-free-li">Asset Free</li>
<li class = "clearfix asset-free-li">Asset Free</li>
</ul>
<ul id = "assetSelectedList" class = "clearfix list"></ul>


![Безумие обратных вызовов в javascript [JS]](https://i.imgur.com/WsjO6zJb.png)


Использование может исключить .asset-paid-li с помощью селектора :not(). Также используйте .clone() вместо получения HTML элементов.
$("input:checkbox").click(function(){
if (this.checked)
$("ul#mosaic li:not(.asset-paid-li)").clone().appendTo("ul#assetSelectedList");
});<script src = "https://ajax.googleapis.com/ajax/libs/jquery/2.1.1/jquery.min.js"></script>
<input type = "checkbox" name = "thing" value = "valuable" id = "thing1">
<ul id = "mosaic" class = "clearfix list">
<li class = "clearfix asset-free-li">Asset Free</li>
<li class = "clearfix asset-free-li">Asset Free</li>
<li class = "clearfix asset-paid-li">ASSET PAID</li>
<li class = "clearfix asset-free-li">Asset Free</li>
<li class = "clearfix asset-free-li">Asset Free</li>
<li class = "clearfix asset-paid-li">ASSET PAID</li>
<li class = "clearfix asset-free-li">Asset Free</li>
<li class = "clearfix asset-free-li">Asset Free</li>
</ul>
<ul id = "assetSelectedList" class = "clearfix list"></ul>