Exploiting a Diophantine approximation of $\pi^4$ into giving a series of rationals for $\pi^4$

Neat Question! What follows is mostly a long comment.

Define $\tilde{\sigma}_s(n):=\sum_{d|n}(-1)^{d-1}d^s$, an "alternating sum of divisors function." Your sum becomes $\sum_{r=1}^x(-1)^{r+1}\tilde{\sigma}_3(r)$. If you now take the classical sum-of-divisors $\sigma_s(n)$, then it's easy to see that $\tilde{\sigma}_s(2k+1)=\sigma_s(2k+1)$ and $\tilde{\sigma}_s(2k)=\sigma_s(2k)-2^{s+1}\sigma_s(k)$.

Your sum is almost the famous Ramanujan Eisenstein sum $Q(q):=1+240\sum_{r=1}^\infty\sigma_3(r)q^r=1+240\sum_{r=1}^\infty \frac{r^3q^r}{1-q^r}$, the difference being your sum is finite and with the caveat that this sum blows up for $|q|=1$. That's unfortunate because for $|q|<1$, there are a slew of identites that $Q$ satisfies, and I'm not sure they will carry over easily in the finite case. It looks like this might be relevant paper, where you'll find a compendium of identities for $\tilde{\sigma}_3(n)$, along with analogous infinite Ramanujan Eisenstein sums:

Convolution Sums of some functions on divisors -- Hahn