Right, as has been mentioned before Lunar stealth is more about hiding in plain sight than just becoming straight up invisible like a Solar would. One of the Herd and Clover Can't be Found are useful for these things. There are also a few Lunar charms that make a target forget something, but I'll need to check when I'm a home and can look at my books to see if they can be applied to stealth. If they can, though, it'd be hilarious watching you Obi-Wan your way past every guard in a palace.
I'll also have to check Clover Can't Be Found, to see if it applies when someone is looking specifically for the Lunar or if it works when someone is simply looking for "an intruder".
There are a few hearthstones that can help a stealth/ninja Lunar, like the Gemstone of Shadows, which lets you cover yourself in, well, shadows. As long as you're in a dark place, it can make you invisible. The Orb of the Unnoticed Predator also makes it so you don't leave tracks, making it harder for someone to chase you. The Gem of Youthful Suppleness can help with the ninja acrobatics part, since it gives you a +2 to Athletics and another +2 to escape bondage (not the kink, more like just any form of physical retraint).
There's a three-dot First Age artifact called an Infinite Resplendence Amulet, which has different variants for every magical material except Soulsteel. At it's base, the artifact lets you change your clothing into whatever you want, provides a couple of ability bonuses I can't remember right now, and gives you +1 bashing and lethal natural soak.
The Moonsilver version of the artifact is particularly useful because it turns that +1 into a much more meaningful +5 natural soak and it gives you +2 to Stealth and disguise rolls. And since it can let you change your clothing for free reflexively, it's like half of Changing Plumage Mastery for any human shapes you have. If you can spring for a four-dot version of the artifact, you could get one that incorporates Starmetal as well, giving you the advantages of both materials. The Starmetal version gives you Elsewhere pockets, letting you stash anything as long as it can fit in your pocket.
And yes, the Changing Moon anima power can be useful because it lets you take basically any form without the restriction of needing to hunt for it first, but this is offset by the fact that if you ever spend peripheral Essence while it's active then your disguise drops.
Just be aware that as a Lunar you can take advantage of basically any timeskip that lasts a few hours to go hunting for more animal shapes (and human shapes too, though those can be obviously more troublesome). So you could use your starting Heart's Blood to start with a few human shapes, and then grab more animal shapes during in-game timeskips, though these would naturally be limited by geography. You can't hunt a Siaka in the middle of a city.