Off the top of my head, your best bet would be to call Spokes near the LA River which isn't that far if you stick to the side streets. You'd probably have to get an Uber or Lyft, but from that location you can ride up and down the LA River which connects to Griffith Park. They have two 8 speed road bikes to rent, but I'm not sure what sizes they have.
https://www.spokebicyclecafe.com/bicycle-rentals
You can also try calling Spokes N Stuff which is the bike rental spot in Griffith Park. I know the owner is a hardcore roadie, because he helps host a morning road ride on Sunday up La Tuna Canyon and back. Also, I'm not sure if he has road bikes to rent, but give him a call.
https://www.yelp.com/biz/spokes-n-stuff-los-angeles-3