It sounds like you want the 5'er more maybe? Save the 4 stringer build for later. I think, as you even said, you would miss the low B.
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another alternative to the SUB might be an Ibanez ATK.
they are available in 4 and 5 string versions. My ATK 300 made in Korea is a bass I definitely regret getting rid of.
Please tell me what building a 4 string Jazz or buying a 5 string Sterling have anything to do with each other

So, now I'm probably looking at putting my own jazz together, I'm thinking I can get in the ballpark, it will sound like a jazz, and it will be completely my own.
Unless someone's giving you the parts, this is a terrible idea and the worst option.
If you actually have to source your parts, it will be more work and money than you expect and the result will have no resale value when you're done.
You can buy a better bass than you can build for the money, especially if you shop hard and/or buy used.
Totally misses the point of building a parts bass and incorrect advice, to boot.