Density of multiples
For computable functions LB from {1,2,3,...} to itself, if

1/(LB(1)) + 1/(LB(2)) + 1/(LB(3)) + 1/(LB(4)) + 1/(LB(5)) + ... ​ computably converges
and
there is a positive integer m such that for all integers i, if ​ m &lt; i ​ then ​ LB(i) ≤ ni

then the density exists and is computable by using m and the convergence to upper-bound the

probability of being divisible by something beyond the cutoff point. ​ (To verify that that argument works despite density not being countably additive, do that for each {1,2,...,n} separately.)

I only have a vague suggestion for possibly speeding up the finite part:

Perhaps in most cases you can find a good-enough bound on the

sizes of most of the high-order inclusion-exclusion intersections.