Start crossfeeding ASAP to prevent a fuel imbalance if you get bogged down further along in the checklists.
What if the engine failure is from fuel contamination?
Will a fuel imbalance become that big a deal that quickly?
I'm thinking priorities here. Fly the plane, deal with the engine failure, then tidy up the rest afterwards, if they're not already covered by checklists.
Reference the original question, airlines may have differing views on A/T use with an engine out. In your scenario, as the engine failure has occurred AFTER thrust reduction to CLB Thrust, you will have to reselect TOGA (either manual or A/T) for the cleanup. Once cleaned up, use Max Cont Thrust.