The book has no idea if you are arbitrage betting since they don't know if you have a second book. The rule makes no sense. If the book doesn't want your action, then they should pay the player and ban him.
There’s problem is bookmaker usually have connections to each other that’s why they can cross examine accounts that doing arbitrage even from different casino.
Bookie connection is the only why they caught arbitrage betting or else you’re right that they don’t have any idea about arbitrage betting.
They will not care if you are doing arbitrage because they are not losing rather the other books in every successful arbitrage.