Having no other purpose, other than to carry it in public, is unfortunately not going to garner much support- even from many in the gun community. In the realm of influencing public opinion, carrying it 'because you can' is usually percieved as a touch nutty by the general public. Their aversion to this display isn't what either local law enforcement or many from the gun right community are aching to contend with. As you should be aware, our legislature is considering a bill to ban the public display of long guns.
I think it would be helpful if you augmented your outing with some significant purpose to your open carry activity. SaveOurState only carries while walking to the range. I would say that it would be wise to minimally, have this as your purpose of carrying a rifle in public. Alternatively, you might consider making this a clear 1st amendment statement by fixing your bayonet to your rifle with a pro-liberty message affixed to it. The rifle then, is also a part of your free speech demonstration.
I cannot comment on what LA county ordinances might exist to restrict this activity. I would refer you to the county website to search their ordinances thoroughly or to contact a competent gun law attorney for advice.