I had the same questions as you had. The M Lift 33 stretches longer than most compact bows you see in the market. I believe you need at least 40" of clearance.
I had a soft case before, blackout 1.5 and it will fit this bow just fine, however soft cases when hunting can hurt your bow. Your transport may be crammed with other equipment and the last thing you need is a damaged CAM before a big hunt.
I decided to go with a hard case, and the one I found is a little more compact than a full box like the V800 or V730.
If you wanted the full box:
The Pelican V730 or V800 works great, it is heavy duty and it comes with a price tag. You may need to remove your stabilizer to make it work. Mine is 6" and it fits just fine.
Blackout also has an option that will work, but at that price tag you might as well just go with a Pelican Vault.
(Cabelas)
I decided to go with a Plano SE44. It went on sale, $89 CAD. (Cabelas)
I also have a Plano compact bow case for my Bear Approach HC, and it works well enough for my needs.
Pros: It works.
1) Not expensive
2) It provides hardshell protection, which is well acceptable for transporting behind an ATV, TRUCK, or Side By Side.
3) Depending on the quiver you went with, mine remains attached to the bow, no issues at all.
Cons:
1) I would not use this for air travel. I would go with a pelican for that or the blackout hard cases.
2) It is not weatherproof. It can handle light rain just fine but don't expect it to be super dust-tight, or heavy rain proof.