Per comments, you would also like https://example.com/forecast/ to be indexed.
You are correct about your assessment that neither a 301 or a 302 fits your use case. Although the 302 resolves the issue of caching, and is a little closer to your use case.
Luis is correct, the 301 is the redirect that passes the value of the link through to the next page a 302 does not. But this passing of the value of the link is based on the search engine checking the 301 several times over several different time periods. They are checking that the 301 remains pointing to the same page, and if not undoing the link effect. So if the 301 is changing there is not going to be the effect as if that page existed with links to all the forecast pages.
Passing the value of the link and user expectations
The value of the link may be limited if you already have links pointing towards the achieve of forecasts. The algo is not a raw count of links where value = links * factor. In other words the link from the HTTP://example.com/forecast to HTTP://example.com/forecast/december-2022 may not have that much effect, assuming /december-2022 is in the sitemap.xml, has other links on the site pointing towards it.
As a general theme for the forecasts, a basic silo structure where the link can be found in a drop-down or sidebar to on December 2022 to other forecasts and other forecasts have the same drop-down or sidebar linking back to december-2022. with a breadcrumb to the main page of forecast ... helps in building the idea that these forecast pages are almost a site in them selves and should be ranked accordingly.
The term silo was coined for structurally building what could stand alone as a site and as far as links are concerned it resembles a site like sites.google.com/forecast but is incorporated into a larger site ... like sites.google.com/forecast. But of course, it links to the main site sharing search engine importance to the main site.
Right now the sub-site or silo has no home page, which is kind of distressing, and addressing that may greatly help your efforts.
If the redirects are not the right tool then what can you do?
To get the effect / results I think you are describing. Build the forecast page ... If the forecast page is nothing more than the current forecast, (which is how your site is currently redirecting traffic), then the canonical tag can be used to specify that the december-2022 should be considered the origin and the forecast page is a copy. The canonical tag is the tool that resolves duplicate content, and you have a use case.
With content on the forecast page it can be indexed and if breadcrumbs are used, and other areas on the site pointing to the forecast page it becomes the home page of the forecast silo, (sub-site), and should rank.
It would be better if the forecast page had unique content and did not have the canonical tag; Although the canonical tag does not decide which page will be listed and the forecast page has the greatest apparent importance. But as it changes the archives are the originals that should be indexed. And well its one line of code to implement in PHP to duplicate the content.
// this code duplicates the ./december-2022/index.html onto this page.
// this is a trusted page on our site, if you can not trust yourself then ...
// as a trusted page there are no security concerns.
// This code has more comments than code.
The canonical tag would be on the december-2022/index.html page.
december-2022/index.htm or index.php
A index.php could include the sidebar or archive dropdown as a trusted include ... although an index.html could fetch the sidebar or archive ... the better choice depends on how large the forecast page is as either the index.html and or sidebar.html could be cached ... the best is what is fastest for the end user?