I have to cover the bullseye with the dot on my front sights for a few of my handguns; specifically, the P365, P938 and G30SF. The first two came with tritium sights, and I added Trijicon tritiums to the last of the three. It seems that they are designed so that you can align the glowing dots to hit POI where the front sight dot is on the target if you shoot in low light. Your LCP Max might just be set up with a similar aiming convention in mind, even if it doesn't have night sights.
As far as your hitting to the left, I would have someone else shoot your pistol to see if that person gets the same results, or rest it on a bench. That way, you can make sure it isn't you. Personally, I have had to kick the sights to the right with more than a few of my pistols to get the most precise POI. Either I've had some consistently off sights from the factory, or something about the way my eye focuses affects things. I shoot with both eyes open, and I've noticed that I might be a bit left if I close my left eye with my Winchester. Different weapons' sights interact with the eye in different ways.