Rodrigo Roesch:
The best thing is to align the images by hand in PS. Yo can then use the lens correction to slightly expand the short exposure to match the size or use free transformation to match the size.
Then make sure the short exposure is on top of the long exposure. Create a layer mask next to the short exposure, select the long exposure, copy it to the layer mask and blur the mask for about 20 pix. You can also adjust the curves on the layer mask to your taste. Once this is done, marge the layers and this will be your new long exposure and you can continue with the next short exposure. You can also use the mask refine function to smooth your mask.
This process is similar to Orion nebula process where you take different exposure so the trapezium does not get blown.
You can see my picture of the eclipse for reference. It is composed by 7 exposures.
Hope this helps
Rodrigo
Could you expand on the idea of using lens correction to match exposures? I haven't found relevant how to's in my searches and I feel like it might be a good candidate for my alignment issues.
When I stack images that were taken very close together (using layer difference in Photoshop), simply moving the images into alignment looks pretty good.

But when the images are at opposite ends of the sequence, taken minutes apart, things are off even after I overlap them as much as possible.

Are lens corrections a good candidate to help here or is it wrong to even try stacking untracked images taken far apart?