Hmm, I've never seen the LCM defined for rationals like that. Anyway, you seem to have provided all the info necessary for a definition and its implications, and what I come up with is:

Suppose we have two positive rationals a'/b' and c'/d'. First reduce to lowest terms to get a/b and c/d. Then let e=lcm(b,d). Multiply both rationals by e to get integers f=ae/b and g=ce/d. The let h=lcm(f,g). Then the final answer is h/e.

I haven't tested this thoroughly or proven it, so if someone sees a problem with it, please say so. It "seems right" for the time being.