I was a lefty pitcher myself. On one hand, I find it basically discriminatory that a RHP can fake to 3rd, throw to 1st, but a LHP cant fake to 1st and throw to 3rd. Its the exact. same. move. Just lefty vs. righty. why wouldnt you be allowed to do it? or why should a LHP have to step off first when a RHP doesnt have to.
On the other hand. as previously mentioned, the whole point of a RHP making this move is to get the guy at 1st leaving on first move. A LHP probably isnt going to get the guy at 3rd because, well, he probably isnt stealing home on first move (There is a slim chance that in your fake to 1st, you catch R1 dead in the water, run at him to engage a rundown, and turn and fire home if R3 makes a move... slight chance i suppose). It would be nice to have the option. I guess the overriding rule here is that you cant fake to 1st when engaged at all period.
I got plenty of runners at 1B in my day with just a straight up good move...unfortunately, i even faked out my first baseman a couple times. Us lefties are at a big enough advantage with that alone i suppose.