For anyone else that may have this issue, I managed to figure it out! Turns out it was a mix of user error and iffy hardware. To get the slime cores, you have to fire your Hyper Beam attack at the slime for a little bit. This isn't an attack you get later, but something you can use from the get go. Hold down the right mouse button to aim, and then hold the left mouse button to fire. Holding the left mouse button first or on its own does not activate the attack, and it's difficult to do on a laptop touchpad (at least for me), which is why I couldn't figure it out. Using an actual mouse, even a blue-tooth one, makes it easy. I imagine I'll be the only one that had this specific issue, but just in case someone else has this problem in the future, this comment will hopefully be helpful.