Hosting it yourself does not give you any extra features. You still need to pay (the same price) for the premium features. The only benefit to self hosting is you have complete control of your data (which is important to some companies)

Reading through the comments on a few of these recent Gitlab posts, the self-hosted option has pricing parity with the SaaS option. I.e. most of the features gated to the $19/month Premium tier for the SaaS are similarly gated in the self-hosted option, and require paying the same $19/month cost in order to activate that extended feature set in self-hosted instances.

There do appear to be some features in the paid SaaS options that are available in the free/core self-hosted software, but I can't find a straightforward breakdown of what those all are.

We're hosting our own instance but we need a feature that is not available in the free version

Self-hosted costs are the same. Bronze is gitlab.com tier, Starter is self-hosted tier.