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I noticed that ってなわけで is commonly used as a substitute for というわけで in colloquial speeches, but what exactly is the purpose of the な-particle in that phrase? Does it make the context that the phrase refers to adjective-ish, like suggested in this answer?
Or is it actually the な-particle that is similar in usage to the one in それなのに? How does the construction work though?
The crucial part you are asking is ってな. As you suggest, it is a colloquial version of という that introduces an appositive clause. They are probably interchangable. The noun is not limited to わけ, but has to be able to take an appositive clause.
という話, ということ, というわけ, という理由, という記事, という理屈, という説明, という計画, という作戦
ってな話, ってなこと, ってなわけ, ってな理由, ってな記事, ってな理屈, ってな説明, ってな計画, ってな作戦
Furthermore, って is a colloquial version of と that introduces a subordinate clause.
...と思う
...って思う
It can also be a colloquial version of the topic particle は.
あの人はかっこいい。
あの人ってかっこいい。
な is probably related to the な that introdoces a na-adjective in attributive position. But that does not explain why it is used together with って. Somehow, とな is ungrammatical. Maybe you can just consider the whole ってな as a fixed expression.
てなわけで. More accurately, the expression you are asking is probabalyってなわけで.