Is the relationship between these two sequences, identical but for signs, trivial? Sums of divisors of product of first $n$ primes has same sequence as products of totients of first $n$ primes, but with alternating signs. OEIS A005867
Mathematica code shown:

Table[Sum[
    MoebiusMu[d] d, {d, Divisors[Product[Prime[j], {j, 1, n}]]}], {n, 1,
     8}]
  Table[Product[EulerPhi[Prime[j]], {j, 1, n}], {n, 1, 8}]

{-1, 2, -8, 48, -480, 5760, -92160, 1658880}
{1, 2, 8, 48, 480, 5760, 92160, 1658880}

Edit an alternate for the first statement:
$$\sum_{n=1}\ \sum_{(d_{1},\dots,d_{2^{n}})\ |\ \text{lcm}(p_{1},\dots,p_{n})}\mu(d_{m})d_{m}$$
Is this a trivial relationship?

$$\prod_{i=1}^n\phi(p_i)=\phi(\#_n),$$
where $\#_n:=\prod_{i=1}^np_i$, by multiplicity. 
$$\sum_{d\  | \  \#_n}\mu(d)d$$
is your second sum. 
Conjecture: for any $n$ that is a product of distinct, single power primes (e.g. $35=5\cdot 7$):
$$\sum_{d|n}\mu(d)d=(-1)^k\phi(n),$$ 
where $k$ denotes the number of $prime$ divisors of $n$, so for $n=35=5\cdot 7$, $k=2$. 
Notice that $g(d)=d$ is multiplicative, so it suffices to establish the conjecture for $n=p$ prime:
$$\sum_{d|p}\mu(d)d=\mu(1)+\mu(p)p=1-p=-\phi(p).$$
Since $\phi(n)=\prod_{p|n} \phi(p)$ (remember $n$ has only single prime powers), so 
$$\sum_{d|n}\mu(d)d=\phi(n)(-1)^k$$
So your observation has been proven.
Fun fact, for any integer $n$:
$\phi(n)=\sum_{d|n}\mu(n/d)d=\sum_{d|n}\mu(d)n/d,$
So you have kind of a mirror identity in your case. Neat!
