As both are replacement effects, I should be able to choose which happens, shouldn't I?
Asking for a friend.
Both dredge and Underrealm Lich's ability are replacement effects. As such, you can choose which one you want to apply when you would draw a card.
616.1. If two or more replacement and/or prevention effects are attempting to modify the way an event affects an object or player, the affected object’s controller (or its owner if it has no controller) or the affected player chooses one to apply [...]