The soul lantern has a relevant 3 frame texture (like the normal lantern), but its animation definition is missing from the flipbook_textures.json file, so isn't being animated.
To verify:
Download and extract the template resource pack: https://aka.ms/resourcepacktemplate
Open textures/flipbook_textures.json in notepad
Search for "lantern"
Note the lack of a "soul_lantern" definition
Still an issue as of 1.21.40.22-preview.