書く方が難しい or 書くことの方が難しい is the way to go.
書くの方が難しい is 100% ungrammatical. As I said in the linked question, 書くのの方が難しい is not 100% incorrect but it sounds fairly awkward at least to my ears. I suppose that heritage Japanese speaker felt the same way I did, but made a simple mistake while correcting it.
That の is a nominalizer in the sense that it plays the grammatical function of turning the verb phrase 一人の相手を幸せにする into a noun phrase, just like the の in, say, 本を読むのが好きです. However, the resulting noun phrase seems to have a more specific meaning than 一人の相手を幸せにすること, which refers to the act of making one person happy in general and abstract terms.
Yes, this の is a nominalizer.
Yes, this の can be replaced with こと.
So is this person talking about how to do 二股? Please always explain the context when you present a sentence based on an unusual context. Also, note that 一人の相手 here is "one girl", not "one of your partners" (see this). Translating 相手 as "partner" is often ...