He doesn't have to drop it, but this is a dip ultimately for him it seems like. So if he takes warlock to 3, he needs to decide which invocation is more important for his ultimate vision for the PC.
Even if you don't do Book of Ancient Secrets, Pact of the Tome will still get you 3 cantrips from any class which is awesome.
Guidance, Spare the Dying (if you don't have a cleric in the group), + another damaging cantrip or mending or something. Pact of the Tome gives better utility than the Magic initiate feat almost since you get a 3rd cantrip vs. 2 cantrips and a 1/Long Rest 1st level spell with the Feat.
Basically Pact of the Tome = a Feat (3 cantrips vs. 2 + 1 1st). Book of Ancient Secrets = Feat+ (Ritual spells from all classes vs. Ritual spells from a specific class). So at 3rd level Warlock you can both or just one and keep your more combat/darkness related ones. Depends on what you want to do.
I probably wouldn't stick with Shadowmonk past 6 anyway if I'm considering a multiclass, unless you really want more ki or speed or unarmed strike damage. I'd probably go Shadowmonk 6 and then Warlock 3. if your game continues past that, you could go Ranger after that! Or just keep advancing as a Warlock. You don't HAVE to take spells that require attacks or saves!