Is it safe to enter the Strip and do its quests without being given the mark of Caesar?

I realise I'll have to stay away from the Topps, but I want to keep the Legion at neutral as long as possible.
So I was wondering if it would be safe to do the quests at the Ultra Luxe, Gomorrah etc?

Do I have to stay away from the Lucky 38 as well?

You just not have to talk to Benny. After you talk to him and then Yes Man, you will be given the mark.

But otherwise, unless you are specifically going against Legion Targets, the legion doesn't care what happens on the Strip.