Regardless of form, I want to know how I can get some interpretation (closed formula preferred) of $B_n$ that is equivalent to $m$ digits of precision, and I'd like to know the fastest way to do it (in terms of calculations required to get the estimate). Can someone please help?
For your information, you may want to consider Wikipedia's page on Bernoulli numbers. I've been attempting to use the "explicit definition" that they give, but I've been running into problems. I'll keep you informed of the progress that I make...
I will assume that $m$ is small. Then a good approximation is obtained from the following $$ B_{2n} = (-1)^{n-1} 2 \frac{(2n)!}{(2\pi)^{2n}} \zeta(2n) = (-1)^{n-1} 2 \frac{(2n)!}{(2\pi)^{2n}} \prod_{i \ge 1} \left( 1 - \frac{1}{p_i^{2n}} \right)^{-1} $$
To obtain sufficient approximation you truncate the product over primes accordingly. See these slides from W. Stein.