UPDATE: Solution was to use the wonderful custom button card and also to use proper padding syntax.
Example:
Note the bottom section of each image. Is there a way to force Lovelace to not stretch them vertically, or, if it does, to keep filling the space with the image by cutting the sides and zooming in?
This is a vertical stack of two horizontal stacks of picture entities.
I really, desperately want my Lovelace UI to look good on my desktop, tablet, and phone. And Lovelace is not making this easy. If someone has a suggestion on a card from HACS that works better, let me know.