Yes, it looks like you have a number of problems.
A. Batteries of this type are not well suited to this type of service, particularly in the way you seem to be using them. They're not meant to be discharged very far.
B. This type of controller has a number of problems. My guess is that your battery is chronically under charged. Don't think that just because it switches off at the higher voltage that the battery is fully charged. More likely, 80-90%. It takes a lot of cycles like this to bring them fully up.
This type of controller also wastes a lot of energy in that the power your panels could be generating during the off time is being lost.
C. It seems you actually have more like a total panel power of 75W using the convential rating system. I doubt you will ever see 120W.
D. Actually, I think your inverter is doing you a favor by cutting off at 11V. Bringing the battery down even this far routinely will significantly shorten the life, particularly for this type of battery.
E. I think you're pretty optimistic about how much energy you're going to get from those panels unless you live in Arizona during the summer.
I did a three-year study of my panels when they were new and found that I only had a 20% (of rated power) utilization factor, mostly due to always having some clouds around and the fact that panel ratings lie.
#1. I suspect you may see an average of 20Ah/day for those panels. If your daily usage is more than 18Ah (because of battery charge efficiency), you will end up in the hole leaving the battery under charged which will kill it.
#2. Adding a second battery will just compound the problem. Your self-discharge will double. Now you will have two more-under charged batteries.
Your basic problem is not having enough charge capability for your usage. This coupled with an inefficient controller is shooting you down.
#3. MPPT will help somewhat if you live in a cool climate. Here, in the tropics, it's not worth the trouble or expense.