For the second part of this sentence, to get the meaning of it, I think we should focus on the whole term "maintain his invincible front" instead of cutting the verb out.
[emoji]Here're some ideas that might be helpful:
1.The word "front" can mean the outward appearence of a person, the way you want to show yourself to others.
[emoji]Eg."That smile is just a front—I don't think she actually likes me at all."
[emoji]Eg."Don't front!---Don't act like a badass when you're not!"
The first example is from Merriam-Webster Distionary and the second one is from Urban Dictionary. Not like these two, the sentence you post is apparently using "front" in a positive way.
So, I would say "He always maintains an invincible front" is just a fancy way of simply saying "He's invincible".
Let me end my speaking with another example from TED[emoji]:
"I was 28, and like so many of us when we are in good health, I felt like I was invincible."
I'm open to new ideas;)
Have a nice day!