If you don't have them already, I'd recommend a sealant syringe with a thin "needle" that can insert all of the way in to the tire via the valve stem. They're about 10 bucks on Amazon -
https://www.amazon.com/Tubeless-Seal...dp/B07WR9MMDW/.
You may also need to replace your valve stems, though. The lightweight, machined aluminum stems will almost always pass the needle; I use Muc-Off because they're available locally. These simplify and speed up sealant maintenance significantly.
That out of the way, I'd just remove the sealant before storing and then re-inject fresh sealant before riding, otherwise you may get a thick, dried up skin where the sealant pooled. This probably isn't problematic for the volume of sealant typically used in a road tire, but my gravel tires have had rather unsightly crusts in them after long off-seasons.