You can use a proper filter. I've been doing so for a slightly different use case (in the bottom of my espresso filter basket to prevent fines clogging the filter, making up for a somewhat subpar grinder).
Of course, you won't find many filters for this purpose, but you can make do with a round paper filter of roughly the size of the basket. For espresso, I use AeroPress filters which I soak under the sink before applying them to the filter basket (before adding grounds).
In your moka pot case, I'd also suggest wetting the paper and applying it on top of the grounds. Given that the grounds are a bit finer than you prefer, I woud use a bit less grounds and push the paper filter onto the grounds before screwing the top on.
The paper filter should ensure no grounds make it into the drink but it may impact also impact the mouth feel of the drink. When I make espresso with an added paper I still get crema, but I'm not sure how much of the oils will get through without the espresso-level of pressure.
Edit: I see another answer also suggested this and focuses on the danger of clogging up. I think that's a valid concern given the finer consistency. If that's something you're really worried about, it's easy to solve by mixing in something coarser that doesn't affect your drink. That way, contrary to mixing in coarser coffee (which you may not have, and may lead to uneven extraction), you could add some inert grainy stuff. I'd suggest some sort of sand, but you'd have to be sure that it's food safe. By goolging, I've found that there are some food-grade sands out there and if you have that on hand it might be an interesting experiment.