Answer: 384

Small Hint:

Let $2u$ be the central angle subtended by each side of length $81$, and set $t=2\cos u$

Big Hint:

Express $\frac{\sin(5u)}{\sin u}$ and the three diagonal ratios in terms of $t$

Solution:

Let $2u$ be the central angle subtended by each $81$-side, and put $t=2\cos u.$ The remaining arc has half-angle $\pi-5u,$ so the chord ratio gives $$\begin{aligned}\frac{31}{81}&=\frac{\sin5u}{\sin u}\\&=t^4-3t^2+1.\end{aligned}$$ This yields $t^2=\frac{25}{9}$ or $\frac{2}{9}.$ Because $5u<\pi,$ we have $t>2\cos36^\circ,$ so $t=\frac{5}{3}.$

The three diagonals from $A$ subtend the same minor angles as $2u,$ $3u,$ and $4u.$ Relative to an $81$-side, their length ratios are $$\begin{aligned}\frac{\sin2u}{\sin u}&=t,\\\frac{\sin3u}{\sin u}&=t^2-1,\\\frac{\sin4u}{\sin u}&=t^3-2t.\end{aligned}$$ Their sum is therefore $$\begin{aligned}81(t^3+t^2-t-1)&=81\left(\frac{128}{27}\right)\\&=384.\end{aligned}$$