[Dilettante_Pro]

М.б. так?

<div onchange="javascript:setTimeout('__doPostBack(\'ctl00$MainContent$fvTrip$ddlCarNums\',\'\')', 0)"
    id="MainContent_fvTrip_ddlCarNums" style="display:inline;">
    <table id="MainContent_fvTrip_ddlCarNums_Table" class="ajax__combobox_inputcontainer" cellspacing="0"
        cellpadding="0"
        style="border-width:0px;border-style:None;border-collapse:collapse;display:inline;position:relative;top:5px;">
        <tbody>
            <tr>
                <td class="ajax__combobox_textboxcontainer"><input name="ctl00$MainContent$fvTrip$ddlCarNums$TextBox"
                        type="text" value="(не выбран)" id="MainContent_fvTrip_ddlCarNums_TextBox" autocomplete="off"
                        style="margin: 0px;"></td>
                <td class="ajax__combobox_buttoncontainer"><button id="MainContent_fvTrip_ddlCarNums_Button"
                        type="button"
                        style="visibility: visible; height: 19px; width: 19px; margin: 0px; padding: 0px;"></button>
                </td>
            </tr>
        </tbody>
    </table><input type="hidden" name="ctl00$MainContent$fvTrip$ddlCarNums$HiddenField"
        id="MainContent_fvTrip_ddlCarNums_HiddenField" value="0">
</div>


<div class="">
    <ul id="MainContent_fvTrip_ddlCarNums_OptionList" class="ajax__combobox_itemlist"
        style="display: none;  visibility: hidden; z-index: 10000; overflow: hidden; width: 178px; position: absolute; height: 368px;">
        <li>(не выбран)</li>
        <li>А161МН177</li>
        <li>В136НО799</li>
        <li>В559НА67</li>
        <li>В599НР750</li>
        <li>Е599НР750</li>
        <li>О161МН177</li>
        <li>О924МУ197</li>
    </ul>
</div>
<script>
var list = document.getElementById("MainContent_fvTrip_ddlCarNums_OptionList");
document.getElementById("MainContent_fvTrip_ddlCarNums_Button").onclick = function() {
     list.style.display = "block";
     list.style.visibility =  "visible";
}
list.addEventListener('click', function(e) { 
   document.getElementById("MainContent_fvTrip_ddlCarNums_TextBox").value = e.target.textContent;   
   list.style.display = "none";
});
</script>