1. In your favorite image editor, create a canvas with a 2:1 ratio, preferably in the resolution of the equirectangular photos or videos you'll be editing, e.g. 3840 x 1920. If you're using Photoshop, choose Transparent as the background. Otherwise, right-click the background color to make it transparent.
2. Put your logo in the middle of the canvas (centered horizontally and vertically). Save as PNG (to preserve transparency).
3. Transform the image so that the logo is on the bottom of the equirectangular image. There are several ways to do this.
a. An easy way is with the Flexify 2 plugin for Photoshop. If you have it, then just select a Latitude of -90.
b. Another way is with the free panoramic editor Hugin. Import your PNG logo into Hugin. When prompted for camera and lens data, select Equirectangular with HFOV of 360.
On the next screen, select the Move/Drag tab. In Pitch, enter -90 to have the logo at the bottom.
Close the Fast Panorama preview to go back to the main screen. Click the Stitcher tab. Save the file and choose PNG as your [export] format. Click Stitch.
Whether you use Flexify, Hugin or some other program, you'll have a new PNG file that's been transformed for use with equirectangular images.
4. To use the logo, just paste the PNG file over your photo or video, or use it as an overlay. In the example below, I added the logo as a layer to a video (here, I used Magix Movie Edit Pro Plus).
And here is a sample where I used it for a photo:
Here is the result:
The next time you want to add a logo, you just need to use Step 4, unless you want to resize the logo or move it to another part of the target photo.