Solution:

The equations factor as $x^4y^4(x + y) = 810$ and $x^3y^3(x^3 + y^3) = 945.$ With $s = x + y$ and $p = xy,$ using $x^3 + y^3 = s(s^2 - 3p),$ they become $p^4 s = 810$ and $p^3 s\,(s^2 - 3p) = 945.$ Dividing, $$\frac{s^2 - 3p}{p} = \frac{945}{810} = \frac{7}{6}, \qquad \text{so} \qquad 6s^2 = 25p.$$

Substituting $p = \frac{6s^2}{25}$ into $p^4 s = 810$ gives $\left(\frac{6}{25}\right)^4 s^9 = 810,$ so $s^9 = 810 \cdot \frac{390625}{1296} = \frac{1953125}{8},$ which means $s^3 = \frac{125}{2}.$ Then $ps = \frac{6s^3}{25} = 15$ and $p^3 = \frac{216 s^6}{25^3} = \frac{216 \cdot 15625/4}{15625} = 54.$

Finally $$2x^3 + (xy)^3 + 2y^3 = 2(s^3 - 3ps) + p^3 = 2\left(\frac{125}{2} - 45\right) + 54 = 35 + 54 = 89.$$