T.F.
A) Get your chemicals out from under the sink RIGHT NOW!! SERIOUSLY! Walk away from your computer right now and imagine there are no cabinets on your doors and take away anything you wouldn't want him to get (in kitchen AND bathroom) and put them up high somewhere-- pantry high shelf, on top of the fridge, linen closet top shelf, shelf up high in your bedroom closet-- anywhere OUT OF REACH because no matter what type of cabinet lock you use, you can not take the chance of the lock not working one day when he pulls on the cabinet door and you are in another room! If you can't find enough high shelves for all of your chemicals, you have too many chemicals!
B) Why are you still reading? You should be moving your chemicals!! haha!
C) Speak with your landlord-- they may be willing to allow you to drill and be happy that you have child-proofed their cabinets for them for the next tennant- you never know-- they may surprise you and allow it if you ask. Be sure to get the OK in writing.
D) I have 3 kids and have never used cabinet locks. All unsafe chemicals are not in reach. As for pots and pans, my toddler has his own cabinet where I keep the tupperware type stuff and he can go in there and pull stuff out to his heart's content. I put colorful plastic links on the doors of that cabinet and any time he goes for another door, I correct him and redirect him to HIS cabinet. It only took a dozen or so times of physically walking him back over to his own cabinet until he got it.