Levelling a slate floor

Hi guys - a newbie here...
I have a rather unusual problem. My house was originally upper floor only, with the slab for the lower floor slighty sloped for drainage and built upon later without levelling funnily enough. Hence I have 2 downstairs rooms on a slight slope which I want to level, and lay a cork floor. No damp problems as far as I know.

There are slate tiles laid directly on the slab, and incredibly difficult to remove. I wish to level the floor in one room initially, which drops around 70mm from one side of the room to the other (around 3m).

The tiles appear to have been either waxed or polished.

Ceiling height is an issue so I want to level it without losing too much height from the highest point.

Questions:
1. Is a latex based floor levelling compound OK or should I be looking at battening the deeper end?
2. If compound OK - should the same mix be used for the deep end as the shallow end?
3. Should I attempt to remove the wax / polish or do I need to get some sort of diamond tipped angle grinder and key the slate tiles?

Any other advice? I will be renovating all interior walls (which are currently just brick) so no skirting / gyprocking issues to deal with.

Thanks

Dom