In the paper by Terrell 1990, in the Theorem 1 below on the page 471, I would like to derive the formulas for $g(x)$ and $h(x)$ and perhaps also why $\beta(k+2,k+2)$ minimizes that integral given.Why there is no $\sigma^2$ nor in $g$ nor in $h(x)$: $g(x)$ is given as $\frac{15}{16}(1-x^2)^2$ therefore we cannot just assume that it has that specific variance!
APPENDIX
Presumably, if you have access to the full article, you might want to look at the Appendix as the paper claims to provide the requisite proof therein. As I do not have access, I don't know.
Regarding your question about how neither $g$ nor $h$ contain $\sigma$, I should point out that when they say "member of the scale family," they are saying that there is an entire family of distributions parametrized by a scaling factor, say $a > 0$; e.g., $$g(x \mid a) = \frac{15}{16a} \left(1-\frac{x^2}{a^2}\right)^2, \quad |x| \le a,$$ whose relationship with the variance is given by $\sigma^2 = a^2/7$, which allows us to specify an arbitrary width for $g$ with the desired properties. In other words, the author is giving you a "canonical" or "standard" form for these.