Given your requirements, here are some Android apps that can record GPS tracks locally on your device and export them as GPX files. These should be suitable for geotagging your photos without needing to send your data to any servers.

1. GPS Logger for Android: This is a lightweight, battery-efficient GPS logger that logs GPS information into GPX, KML, and/or CSV files. It’s open-source and doesn’t require an internet connection to log data, which should be suitable for your needs.

2. OsmAnd: This is an open-source navigation app that has a built-in trip recording plugin. It can record your GPS tracks and save them as GPX files. The app also functions fully offline, so you don’t need to worry about it trying to “phone home”.

3. My Tracks: This is a simple, no-frills app that logs your GPS tracks and exports them as GPX files. It’s not as feature-rich as some other options, but it’s lightweight and easy to use.

4. GPX Viewer and Recorder: This app allows you to record your path, save your paths and share your position. It can save the data as GPX files which you can then use for your photography geotagging.

All of these apps should be able to run in the background while you’re using your camera, so you can just start them up at the beginning of your hike and then forget about them until you’re ready to export your GPS track. Then, you can use a software tool to add the geotags to your photos based on the GPX file and the timestamps on your photos.