Yes, you will have trouble fitting it in, and likely even with lid removed since it's a hard case and can't be squished momentarily to get it into the bin. It's not so much Southwest's limit, as the height of bins on 737 planes (which is all that Southwest flies). With something tall, you have to squish and turn and maneuver and everything to get it into the bin. I know because I just went through this with a soft case that was overstuffed to a bit over 10" height and it was a bear to get in, and I could only get it in because I could squish it down momentarily (though that was hard because of how overstuffed it was).

Height is the wrong dimension to go over on. Think about how a bin is constructed. Height is narrowest dimension, depth is next, and length is the negotiable one. (A 19" bag will go length-in on a 737 bins, but if it's longer, it's still go depth-in if the airline doesn't stop you from boarding with it.)

But you can't turn a bag that's got too much height in any direction to get it fit in a bin where the only flexibility is length.

That likely only leaves the closet. But that's first-come first served, and if you're on a flight with connecting passengers (common on Southwest, uncommon on "legacy" airlines), the closet may already be occupied. And since you can't have "status" with Southwest, and you're not flying up front (where they give you more flexibility) in a two-class plane, I don't know you'll convince the flight attendants that it's your bag, not any other, that should go in the one closet on the whole plane.

Given all this, I hope you're on the first flight in the morning, and I hope it's a nonstop. Because the first flight in the morning is the least likely to have through passengers, and if you have connections then you'll have to go through the same thing repeatedly!

Is there any possible way you can pack the item (insane amount of bubble wrap surrounding it or whatever) that would "comfortably" fit in 9" height or less, even if it's ugly? For something you're carrying on, it need not be a "suitcase", it can be anything that won't fall apart and will protect the object. (The only thing is, TSA might not let you go through security without "unwrapping" it, depending on what it is, and depending on whether you're a Trusted Traveler who gets PreCheck or have to go through the (comparatively more "unpack this, take this out" rigorous) "normal" security line.)

(You've only mentioned a worry about the item getting lost, not about it get damaged, nor about it being heavy or hard to carry unpackaged, so I'm not clear on how important the packaging is.)