In the TitleBar module is it possible to limit the list of actions to only export results?

Unfortunately not. the TitleBar module takes a parameter called actionsMenuFilter, which sounds promising but cannot do this -- it can only take three values, 'search', 'dashboard' and 'dashboard-SimpleDashboard'.

The first controls the presence and absence of all of the save-search, save-eventtype, and export stuff, all together. The second controls certain stuff like 'schedule this dashboard for printing', and the last one is an internal thing that puts up the 'edit this dashboard' link.

In the future we could and probably should broaden this parameter to allow individual items to be switched on and off. (If you want to increase the likelihood of this happening or speed it up, feel free to send an email to support@splunk.com and describe what you're trying to get done.)

and if you really need this so much that a) you're willing to create and own a custom CSS file, b) you're willing to convert your view to the advanced XML and c) you're OK with it being a green button saying 'Export' ...

then you can put a module into your XML tree called ResultsActionButtons, and use the following custom CSS in that one view to remove all the buttons except for the 'Export' button.

.ResultsActionButtons .saveButton,
.ResultsActionButtons .printButton,
.ResultsActionButtons .linkButton,
.ResultsActionButtons .editButton,
.ResultsActionButtons .splViewEvents {
    display:none;
}

/*
    And as a bonus, we'd like the export button to be green. 
    (Normally it is grey in this module)
*/
.ResultsActionButtons .exportButton {
    background-color:#73A550;
}

A couple people out there in the wild have done this trick, and once the version of the ui_examples app for 4.1 is up on splunkbase, there's a living example of exactly this in there that you can just check out.