The drop-down is very misleading.
Can I not use "Ich hätte gern" here?
Surely "... In meiner Milch"? (dative form)
Well, while implicit, you ask someone to put the sugar into the milk and hence to perform a directed act, and as you remember, then (with direction / movement) we use accusative in German.
Can you use "Klumpen" for lumps?
"Stück(e) Zucker = Zuckerstücke = Würfelzucker" is the more usual German expression in regard to sugar. I have not heard "Klumpen Zucker" in the German language up to now.