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The "1st gen" SE is the same size as the 5, still available from e.g. Amazon for $100–$200, and runs the current iOS very well if a little slowly.

I ran those for many years but unfortunately the batteries are old and seem to only last a year these days

Why not go to an Apple Store, and buy a new battery?

running with 1st gen se since fall 2018: battery is still at 95%

replacement kit from ifixit is available thou