The French Wiktionary is currently using the fixed list approach for languages, administered by the community. We use ISO codes to represent languages almost everywhere (e.g. the title of a language section is {{langue|fr}}, which is displayed as "French"). We do not have identifiers for dialects and we usually use a geographic term. Not sure if those should be as controlled as languages.

I expanded the description above. Here a few precisions and an explanation of a possible solution.

1. The other day we looked into the way that the dialect of Finnish place names should be specified (discussion, WP article on some).
2. It seems that a few may have ISO codes/IETF tags, others don't. In the current way monolingual text codes work, I think one would first have to ask the relevant IETF appointed administrator for these tag to define a variant tag and then wait for WMDE to add them. Alternatively, if we leave the interface open as we do for new item, one could just make up a code on the fly. The later isn't really desirable.
3. To ensure that the variant is identified, one would need to define it through a QID of an item. This as not all variants have IETF-tags or have IETF-tags known to Wikidata.
4. Whenever possible, output should use IETF tags. These could be defined on the item of the language.
5. If no IETF is defined, the item of the language could also offer a fallback. (just "fi" instead of "fi- hämäläismurteet"). Possibly a fallback could also be to "fi-x-QID".
6. Once in a while, we could request new IETF tags for variants that need them.
7. To simplify input by users (and bots/tools), it should be possible to input IETF tags (e.g. "de") instead of a QID.