My first thought is to challenge the assumption that these are iconic features for these classes, and say that they don't need to be reproduced in A5E
but that's not in aid of the question!
To answer: Bard's Loremaster archetype gives something similar but not the same... but if you want to give them something like Magical Secrets, then you need to take something away. Removing hymns effectively neuters the big A5E part of the class.
Really, and this'll probably go for fighter too, IMO they gained a lot more than they lost going from 5e to A5E. But you could scale back the potency of the A5E features for the bard, if you need to give them magical secrets. Make Hymns cost double the bardic inspiration that they do now, or kill some of their skill bonuses.
For fighter... action surge, hm. Make it a fighter-only maneuver that costs all of their exertion, and/or make it so that they can't regain exertion until after a long rest once it's used? Otherwise, maybe trade the maneuver specialization feature for it?
Wanting to have your 5E cake and also eat the A5E features too, while trying to call it balanced- that's a tough nut to crack.