Is it possible to invent a word which would describe rule by the loudest?

Not δυνατότερο. One, because that’s Modern Greek, not Ancient; Two, because Modern Greek doesn’t have a distinct word for “loud”, it just uses the word for “strong”, dynatos. (In fact the OP’s form is “stronger, louder”.)

Actually looking at Woodhouse’s English-Ancient Greek Dictionary, Ancient Greek isn’t much better. The words given for “loud” are literally: big, sharp, clear-sounding, upright, bitter, piercing, over-toned, shining, noisy, roaring (of waves), sonorous. They’re mostly ambiguous, which disqualifies them for me.

Gegōnocracy, “rule of the sonorous” is the least ambiguous as a word, but it would end up ambiguous in English with gegonocracy “rule of facts (what has happened)”; in fact, I wasn’t familiar with the adjective gegōnos.

If I had to pick one, I’d go with rhothiocracy, “rule of those roaring like waves”. I don’t love it, it’s not actually that commonly used of people. But Aristophanes did use it to refer to popular acclamation (albeit in a mariner context): “raise loud waves of applause in his favour this day” (Knights 546)

My preference would be to go with shouting as the root notion here, certainly out-shouting your opposition: “Rule of Shouting”. Boo-cracy (< bo-ē) is ambiguous with the Rule of Oxen (< bo-os), though that may be a feature and not a bug. (EDIT, h/t John Gragson: maybe instead Boēto-cracy “Rule of the Shouty”.)

Craugo-cracy (< krazō, kraugazō) works best for me, and a kraugē is typically an angry, not a joyful shout.