Hi,
I did work with pretty fatty tissue (skin cancer biopsy and pig ears). I did not mount them in ice so my situation would be different from yours.
The only thing I can say is that in my case, I set the cryostat at -30 and I cut the section on the opposite side of where most of the fat was.
Once the slice was mounted either on a tlc foil or on the target plate, I used gentle washing inclining the plate and with the aid of a syringe. Using chloroform is very effective for lipid removal.
I hope this helps a bit
Simona