I thought this was going about the lack of a Drizzle user in LC (Rain Dance?), but since we're on the topic of LC suspects, what about Exeggcute? I run a Harvest Exeggutor with a Lum Berry and Rest on my Trick Room + Sunny Day team, and it's pretty good at stalling, besides that whole 7 weakness thing. Harvest Exeggcute could theoretically heal to full health every time it gets below 50% with an Berry Juice, because of Harvest recycling the berry. That seems pretty cheap, although I'm not sure that it can survive a sucker punch, or any one of its 6 other weaknesses. But hey, if the move leaves you with 5 HP or more, you can heal back to 100% health. I whipped up a quick set despite knowing nothing about Little Cup:
Exeggcute (M) @ Berry Juice Trait: Harvest
EVs: 196 HP / 196 Def / 116 SAtk
Bold Nature (+Def, -Atk)
- Giga Drain
- Psychic
- Sleep Powder
- Hidden Power Fighting/Reflect/Light Screen
This reaches 18 def, 25 HP, and 14 sp att with my EV spread (IDK how do LC EV spreads, I just used pokemon online to get the max possible stats in HP and Def, then put the rest in Sp att). Giga Drain heals you, and that's the only reason I picked it over Energy ball. Psychic is for STAB, and Sleep Powder shuts down something like Zuruggu for a couple turns. You have a couple of options for the fourth move. Hidden Power Fighting is a 2-3HKO on pre evo stone Zuruugu, and that's pretty much the only reason I have it. Although Exeggcute is weak to Crunch, it only does 96% at max, which you can then heal back to full health using berry juice+harvest. You could also run Reflect for even more defense, or light screen to patch up your rather low 11 sp def (at least, it seems low to me).
Note, if Berry Juice isn't a berry, and can't be restored with harvest, that pretty much kills this set. Oran berry could work, although it only heals half as much.
Sorry if I'm posting this in the wrong set, but it seems like it might work.