You could have a wooden cube constructed of the same wood or material that your new vanity is made from. Have a hinged top with padding on it for use as a stool but have a a hole cut on one side for the cat to enter and put the litter box inside. Just lift the seat to clean the litter box. If you kept your current vanity you could put the litter box in the side with no sink. Paint the interior of the cupboard the same as the vanity and cut an opening and finish it off with wood moulding. Your cat will have privacy and the litter box will not be in visitor's site. A third option is to try and toilet train the cat. I am always fascinated with cats that will actually use the toilet!

I have seen a litter box that was entered through a pet door. The actual box (enclosed) was in the next room
, which happened to be the garage!

People we know put a pet door on the front of their vanity cabinet and put the litter in there. This was a double vanity, of course.
If you did this or built a separate cabinet for the litter, you could install a sliding tray that can be locked in place when you are not tending to the litter. When it's time to clean, you can open door and slide out.

Our cats' litter boxes are in a coat closet that we aren't currently using. We just leave the door cracked.

What ever you build or install make sure that it's not too short and that it has good ventilation.

Your plan sounds like a good one - could you not stretch the counter-top wide over the space and place one cabinet below leaving the other space open for the litter box? I don't know your style but you could hang a panel below to hide the kitty litter box. You need easy uncomplicated access to it to clean it out.
I have one that has a cover to it, (like a transportable cake carrier - but with an entry way).
The other thing you need to consider is that inevitably some kitty litter gets kicked out and this is not nice for guests. Placing a rug or mat in front might help. You can pick this up and pour it in the toilet or shake it in the trash.
These are not design solutions per se, but practical considerations from one who has been there and done it!

Great ideas! Thanks everyone.