I don't think you could ever build a fifth floor, only four floors up and four floors down into the basement. You could probably add a really tall foundation and build livable floors on top of that.
Did they add the fourth floor at some point? I watched Jack Septic Eye's Sims 4 series and he was gutted his house could only go up to three levels at the time in 2015.
You could terrain sculpt a tall hill and build your light house on top of it. Seen many images of shorter lighthouses built on top of tall rocky hills and I think they look pretty cool.
Make the lighthouse slimmer, it will look taller. Assuming you already use the tallest rooms. Also raising ground level should raise the build, but I never tested. Like others suggest raising the foundation will help too. To avoid spending the top floor on the light function, see if it would be possible using a lamp object on top, that you of course will enlarge until it's super huge .
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Did they add the fourth floor at some point? I watched Jack Septic Eye's Sims 4 series and he was gutted his house could only go up to three levels at the time in 2015.