Tout du moins is not a correct expression as far as I know. In any case, I have never heard it. À tout le moins, on the other hand, is correct.

There are also these other similar expressions with moins:

• Du moins
• Au moins
• Tout au moins

Both are correct but à tout le moins sounds old-fashioned now (wiktionnaire).

There’s also pour le moins which has the same meaning.

“Tout du moins” seems to me a less common variant of “tout au moins”. For the rest, both expressions mean pretty much the same same thing, roughly “Well, at least, nevertheless, anyway”, though quite more formally than several of these expressions.

Absolutely, “tout du moins” is incorrect. You can also use “pour le moins”. I think “du moins” corresponds to a more relaxed language, which is not the case of “à tout le moins”.

“Tout du moins” is absolutely correct, like “du moins”. “Pour le moins” is perfectly correct too, but sounds very literary & a bit old-fashioned as mentioned earlier.

À tout le moins est en effet plus soigné et littéraire que tout du moins.

Il est vrai que l’expression est un peu recherchée, mais elle est compréhensible sans effort par une oreille francophone