How to align burst photos by fixed features in the pictures

This is one of those things where the images need to be sent to Photoshop. You have at least two options.

In Photoshop, if you choose the command File > Scripts > Load Files into Stack, the Load Stack dialog box offers the option “Attempt to Automatically Align Source Images.” After it does that, you can crop the entire stack to get consistent framing across images. Then use the command File > Export > Layers to Files so that, obviously, you get a new file for each image in the now aligned set.

Another method is to load all of the images into Photoshop yourself as layers in a single Photoshop document. If you’re starting from Lightroom Classic, you can use the command Photo > Edit In > Open as Layers in Photoshop.

Next, choose the command Select > All Layers, and then choose the command Edit > Auto-Align Layers. If you use this method, try the Reposition option, or maybe Collage. The other options are more for merging images into a panorama. (The left four options will attempt to apply a spatial projection to make a panorama look right, and you don’t need that for this).

If for some reason neither of those automatic alignment features work, the last resort in Photoshop is to nudge each layer into alignment manually. This goes much faster if you set an upper layer to Difference blending mode; when two images are perfectly aligned all of the aligned content goes black (only the differences are shown). Then you set the layer back to Normal blending mode and do another layer.

You could try Photo Merge > Panorama, although it's not intended for the purpose you describe. A better option would be to send the Photoshop using the 'Open as layers' option, then manually align the layers.