This is my understanding:
"どうしようもない" is a complete sentence. (There is nothing that can be done.)
"noun + もない" is the sentence structure.
therefore: "どうしよう" = "noun"
"どうする" is a sentence ending with a verb (ie. not a noun).
Is "どうしよう" a noun? Is it the same part of speech as "どうする"? <--- question #1
If "どうする" = [part of speech] = "どうしょう", then "どうしようもない" is not grammatically correct. A verb cannot be the subject.
So, there are 3 ways to change a verb to noun:
(#1) "する事"
(#2) "する物"
(#3) "するのparticle"
"noun+もない" needs a noun as the subject.
"どうする" surely is the subject in "どうしようもない", but "どうする" is a verb.
so, using method #3, we use a particle to change "どうする" into a noun. The particle is "も".
Therefore, is the phrase "どうしようのもない。" grammatically correct?