I'm a bit confused by this. You say that this loop is 9 actions, but wouldn't that only be if you were waiting for Slowcake's Amanuensis' card to show up? In that case, why not just wait for the Urchins' faction card and get favours from it directly (If you are trying to be as action efficient as possible and relying on card pulls anyways)?

Assuming you have high enough BDR you can summon the Amanuensis with an action, then use an action on that card to raise your notability. Once you have done that 2 action loop 7 times (you should be at 8 notability if you want to be able to summon the Amanuensis after), then you can Insist on Favourable Circumstances and gain a favour from the Urchins' card. If I counted correctly, this whole loop takes 16 actions, which means if you are doing this method for Urchins favours, then you would be better off doing the A Boxful of Intrigue carousel in 14 actions.