How can I efficiently convert TEI documents into LaTeX?

It's an XML defined markup. There are XSLT stylesheets for this format from the same website you link to

https://www.tei-c.org/Tools/Stylesheets/

This is a set of XSLT 2.0 specifications to transform TEI XML documents to XHTML, to LaTeX, to XSL Formatting Objects, to OOXML (docx), and to ePub format. The files can be downloaded from the Releases area of https://tei.sf.net . They concentrate on the simpler TEI modules, but adding support for other modules is fairly easy. In the main, the setup has been used on ‘new’ documents, ie reports and web pages that have been authored from scratch, rather than traditional TEI-encoded existing material.