There is no significant difference in meaning between 「[一旦]{いったん}~~」 and 「~~が[最後]{さいご}」. You can use either of the two or even both together to say:
"Once someone does A, he will always (or never) do B." or
"Once A happens, there certainly will come situation B."
In other words, it expresses an automatic and/or uncontrollable result.
I will use your sentence in my explanation.
「あいつはマイクを[握]{にぎ}ったが最後、[離]{はな}そうとしないカラオケ[狂]{きょう}だ。」
↑ Either 「たが」 or 「たら」 is OK.
= "He is such a big karaoke maniac that once he grabs hold of a mike, he will never let go of it."
To say the exact same thing, you can use 一旦 or use both 一旦 and が最後. The sentence will only sound more emphatic if you use both.
「あいつは一旦マイクを握ったら、離そうとしないカラオケ狂だ。」
↑ Only 「たら」 can be used.
「あいつは一旦マイクを握ったら最後、離そうとしないカラオケ狂だ。」
↑ Either 「たら」 or 「たが」 can be used.
「一旦」 can be replaced with 「[一度]{いちど}」 in these constructions. (But 一旦 would sound better to more native speakers.)