Build out both sites with different content and rank them both and take multiple top spots in Serps. If the second way is truly misspelled, then build that name as a brand and don't target the misspelling in your content. Just rank for that keyword as a branded one.
If you don't mind the extra work and time, I think you should definitely target both, at least redirect while you set up both. Or you could build a third website that consolidates both. usually, the more work you have to put in, the better it is in the long run, if that's your plan...
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