Another related question popped up: If I’m playing as Ironheart, am I able to choose not to ready Brawn at…

No, you cannot choose to leave Brawn exhausted at the end of the player phase if nothing is preventing him from readying. You must ready cards at the end of the player phase, unless an ability prevents it or requires an additional cost to be paid to ready the card, in which case you can choose to not pay that cost.