Solution:

The smallest angles $\alpha, \beta$ of the $3$-$4$-$5$ and $5$-$12$-$13$ triangles have $\tan\alpha = \tfrac34$ and $\tan\beta = \tfrac{5}{12}.$ By the tangent addition formula, $$\tan(\alpha + \beta) = \frac{\tfrac34 + \tfrac{5}{12}}{1 - \tfrac34\cdot\tfrac{5}{12}} = \frac{\tfrac{14}{12}}{\tfrac{33}{48}} = \frac{56}{33}.$$

The third smallest angle $\gamma$ satisfies $\gamma = 90^\circ - (\alpha + \beta),$ so $\tan\gamma = \dfrac{33}{56}.$ The right triangle with legs $33$ and $56$ has hypotenuse $\sqrt{33^2 + 56^2} = \sqrt{4225} = 65,$ a primitive triple. Its perimeter is $33 + 56 + 65 = 154.$

Thus, the correct answer is C.