Er, just my two cents:

届かん = 届かない
言わん = 言わない
分からん = 分からない
etc, etc.

It's the negative (though this form of the negative is also used in Kansaiben etc - other than the 〜へん negative form).

Anyway, the grammar structure is '〜とばかりに' （いかにも〜という様子）which can connect to the negative form of the verb. So in the case of 届く, it can be 届かんとばかりに。(It has the same meaning as 〜んばかりに, which connects to this particular negative form of the verb)

So IMO, 『天に届かんとばかりに燃え盛る』would mean something along the lines of "burning brightly/blazing as (if) to reach the heavens"

Hope this helps!

ウチはネイティブじゃないから間違ってたらすんません。どうぞ直して下さい。