Do you have room to keep one quiet switch in the existing closet? Maybe mounted to the ceiling?
Can you run 4 uplink cables from the closet to the basement?
Fanless managed switches with 4x 10G uplink ports and at least 16x 1G access ports are reasonably cheap (minutes transceivers, of course). For example:
https://www.fs.com/products/134655.html?attribute=8032&id=289447
If you can fit the cost into your budget, use CAT8 for your uplinks. That way if you ever do decide to upgrade your access network to 10G, you can bump your 4x uplink up to 25G without having to rewire.
Not that you'll probably ever need that much bandwidth in a house in the next decade or two.