I suspect not.  Here is intuition (not a proof) for why this cannot be done for large n.

Fun prime fact:  if the first k primes are all at most n, then the next k primes are all less than 3n. You can verify this by hand for small k and n, and appeal to a Chebyshev type estimate for the rest. For a positive solution to this problem for large n, there is a b not much smaller than n/2 so that primes in (b,n) determine most of the pairs:  many of the pairs cover a prime in (b,n), and most of the rest cover twice a prime in (b,n/2). By the time you have picked candidates to handle the primes in (b,n), you don't have many options left to pick a new number to cover (some multiple of some) primes in b.

In particular, many of the pairs chosen have no odd multiples of three.  Now you don't need to cover many odd multiples of three, but if you start building a cover to take care of the large primes first, I believe you will run out of options (and your cover will fail) before the time you reach primes the size of n/4.  More specifically, the existence of the type of cover proposed suggest to me too many pairs of consecutive numbers of the form (2p,3q) for p and q prime.

Gerhard "Doesn't Quite Cover It, Unfortunately" Paseman, 2019.04.09.