"Advantages of whatever THAT happens always outweigh the costs"
In the sentence as above let us attempt clause division.
Advantages (of whatever THAT happens) always outweigh the costs"
•Advantages always outweigh the costs"— is the main clause.
••of whatever that happens— subordinate adjective clause
— in which (That) has no validity, whereas (OF) establishes the relationship bonding with "advantages." You may retain( that ), but it that case (of whatever) is rendered dangling a-begging a predicator or, so to say another clause construction. OF, IN, TO, FOR Or for that matter many such preposition before subordinating conjunctions are relation markers. In the last analysis, it is better that we must drop THAT from the sentence.
However if we replace (whatever) with, say, (things) the subject phrase would be(Advantages of the things) of the main clause, and in that case you may retain (that).
By the way, WHATEVER is a relative pronoun / a subordinating conjunction and both the case it joins two clauses and require no other conjunction as THAT.