What method do you use to generate export the static tiddlers?
I quick look around suggests $:/core/templates/exporters/StaticRiver/Content transcludes $:/core/templates/static-tiddler which contains;
<a name=<<currentTiddler>>>
<$transclude tiddler="$:/core/ui/ViewTemplate"/>
</a>
Theoretically if you make an alternate or clone of $:/core/ui/ViewTemplate and point to that instead it will be used to generate the static tiddlers. In your clone you remove the elements you do not want to appear on your static tiddler.
Perhaps changing this
<$list filter="[all[shadows+tiddlers]tag[$:/tags/ViewTemplate]!is[draft]]" variable="listItem">
<$transclude tiddler=<<listItem>>/>
</$list>
to only transude some named tiddlers, not all with the tag[$:/tags/ViewTemplate]
This may be the wrong template to change depending on the method you use to generate static tiddlers.