Can you elaborate more on what your capture settings are? Pro Mode or Auto, or HDR? Or are you using the Long Exposure Tech Preview feature? Any additional details you can share would be helpful.
Is your iPhone SE the 2016 or 2020 model? Are you updated to 14.0, or all the way to 14.3? Have you tried starting your device or reinstalling the app (after you first ensure everything is backed up via iTunes device Backup or synced 100% to Lightroom cloud storage if you are subscribed)?
Regarding the image looking blurry, the zoom in both the native and Lightroom cameras is really just a crop (digitally cutting out part of the original image). The remaining pixels get stretched to fill the screen, and a very high zoom can look blurry as a result. iOS may process their images different than the LR camera, which is why digitally zoomed images may look differently comparing each camera.
Ah, I think you are talking about the quick review mode that shows the last image captured. That is a tiny thumb image compared to the original, and will not look as good as the original that you will see after you exit Camera mode.
As far as the device running hot and draining qiuckly, is this during capture, during editing, during sync?