A bit of trial and error and the solution to my prob is quite easy
Use a custom template in your call, like so:
[!AjaxSearch? &ajaxSearch=`0` &tplLayout=`@FILE:assets/snippets/ajaxSearch/templates/my_custom_template.tpl.html`!]
In my case, my template only kept the form markup:
[+as.showForm:is=`1`:then=`
<form [+as.formId+] action="[+as.formAction+]" method="post">
<fieldset>
<input type="hidden" name="advSearch" value="[+as.advSearch+]" />
<label for="ajaxSearch_input">
<input id="ajaxSearch_input" class="cleardefault" type="text" name="search" value="[+as.inputValue+]"[+as.inputOptions+] />
</label>
<label for="ajaxSearch_submit">
<input id="ajaxSearch_submit" type="submit" name="sub" value="[+as.submitText+]" />
</label>
</fieldset>
</form>
`+]
Thank you, Coroico for this! Very nice!