「～～だろうが」 is an accusatory sentence-ender used primarily by male speakers. The 「が」 is a particle.
「がい」, though not too common, is an emphatic and tougher-sounding version of that 「が」.
Likewise, for emphasis, 「か」 for questioning becomes 「かい」 and 「だ」 for affirmation or declaration becomes 「だい」.
Though 「かい」 and 「だい」 are far more common than 「がい」, I do not think 「がい」 is dialectal. (The speaker in the video grew up in Kanagawa, which is right next to Tokyo.)
