Yes. Quarantine your inverts. I will explain why.
Part of the life cycle of both ich and velvet involves the organism latching onto a hard surface, glass, plastic, snail shell, crab shell, etc, to grow and release the free swimming stage which goes on to infect fish. There's no obvious signs of this, so the only way to be sure is to put them in a system without fish for a minimum of 45 days (Velvet), or 76 days (ich) as that is how long it takes for each parasite to starve to death, being unable to feed on fish. Unfortunately the copper based medicines that would speed up the process would also kill the inverts, hence why it takes so long.
That being said, 'soft' inverts such as urchins and anenomes, ich/velvet cannot latch onto, so these only need a good rinsing or two after acclimation before introducing to the display tank.
That is why we recommend quarantining inverts. I learned this lesson the hard way, when velvet got in my tank aboard a snail I did not quarantine and lost most of my livestock.
You actually only need two tanks. Use one to quarantine the fish, treat with copper or however you want. Because it's only a 10 gallon, it's easy and fast to remove the copper via Cuprisorb and water changes to then dose prazi.
Keep the other 10 gallon as your invert quarantine. Heck, set it up as a nano tank with rock and whatnot since you won't be using any meds on this one. Run a batch of inverts through it, once they go into your display, add the next batch of inverts. Batches are best, but if you can keep track of which inverts were introduced when, at each 76 day mark they can be put in the display. What I mean by that is if you put snails in there Day One then Day 30 you put in hermit crabs, you can take the snails out Day 76, and the hermits Day 106.
I hope this explains and answers your questions.