There are many possibilities. One possibility is to take a pair of families to cover positive and negative excess kurtosis.

The family of scaled t-distributions have parameters ($\sigma$ and $\nu$) that affect the variance and kurtosis. They can only have kurtosis above that of the normal, though.

That will have excess kurtosis $\frac{6}{\nu-4}$ (so if you want that to only go as high as 3, you'll want $\nu\geq 6$).

It has variance $\sigma^2 \frac{\nu}{\nu-2}$, so given $\nu$ you can choose $\sigma$ to yield the desired variance

The family of scaled shifted (to mean 0) beta distributions then would take care of the case where kurtosis was smaller than for the normal (both families include the normal as a limiting case). So take a $\text{Beta}(\alpha,\alpha)$ and shift it down by $\frac{1}{2}$ and then scale to the desired variance.

That is a $\text{Beta}(\alpha,\alpha)$ has excess kurtosis $-\frac{6}{ (2\alpha + 3)}$, and includes your desired uniform at $\alpha=1$.

Before scaling it has variance $\frac{1}{4(2\alpha+1)}$; the ratio of the desired variance to that unscaled variance will be the square of the required scale.