They make turbine vent covers that block the vent on the outside so you don't have to deal with dead animals.
There's solid ones (I guess you could build a little base frame if you want a solid one and it's not tall enough):
Various heights exist if you hunt around:
If none of those fit you can purchase wire mesh and just build it into an appropriately sized box yourself. Wire mesh panels are fairly easy to attach together; there's a lot of options but if you have no special tools some short bolts + washers bigger than the holes will do the job just fine. You can cut heavy mesh with cheap tin snips, or a grinder if you have one. You can fold it on the edge of a table with some thick gloves and a hammer or a piece of scrap wood, or your foot. This stuff for example is regularly sold at big box hardware stores:
If your situation is more complex, you can call local wildlife control and they can evaluate the whole setup and give you control options.
As for screens on the inside I can't think of a problem with that except having to clear out dead animals, which might get kind of gross if you forget to regularly check, or kind of tricky (or potentially dangerous) if you see them while they're still alive.