Since I already explained the optimal solution to the basic question mentioned in the summery lets solve the bonus question too (my solution also matches the solution given in comments on the article side so it should be good (and said to be correct by the author) - note currently no answer with a high score is correct - mine has 1).

The bonus question is that you play two rounds, and your oppoent must play atleast rock once. So, if he plays something not rock in the first round he must play rock in the second and loss (you just play paper). If he plays rock in the first he can play 1/3 all in the second (which leads to a draw like normal). Thus, if he plays rock first it is like normal RPS (because he get 0 in the next). Otherwise you get one free win (for the second round).

Thus, we can model the first game of the bonus question as (where the numbers is the number of rounds he wins on avg given the choice in round 1):

R P S

R 0 -1 1

P 0 -1 -2

S -2 0 -1

Where you pick columns and him rows. We see that rock dominates paper for the row player. We get

R P S

R 0 -1 1

S -2 0 -1

For the column player, the choice of rock now dominates scissors. We get

R P

R 0 -1

S -2 0

Playing rock 1/3 and paper 2/3 for the collumn player gives -2/3 wins on avg. Similarly, the row player can get -2/3 wins on avg by playing rock 2/3 and scissors 1/3.