Do not drill. I did, big mistake. Did not drill deep, and drilling was not the demise of the fridge. I used short sheet metal screws. First two went ok. The third offset screw punctured a line. Had the Company not insisted it be done Right Now, I would research alternative applications. Marine lock made by 3-M company, sells adhesive pads with cable to padlock door shut. Reviews we're all good. Just look up marine lock for fridge. NO HAPPY DRILLING HERE !
This isn't an answer that will solve the problem, but rather provides context. I grew up with this sort of stuff, working with my father as a finishing carpenter until I started my own career. However, the majority of our work was in new houses, so not a lot of renovation jobs.
In my experience, baseboard can either be installed before or after the tile, but if it is installed before, the baseboard is installed at the level the tile will be, leaving a gap below the baseboard to accommodate the tile. Once the tile is installed, quarter-round (or door-stop for a smaller profile) is used to cover the gap. If the baseboard is installed after the tile is installed, it is typically scribed to match the variations between tiles (quarter-round is short enough that it can bend).
Installing the baseboard first is easier because you can avoid the scribing, but if the customer didn't want quarter-round then we scribed it; usually the tile was pretty flat but there was always some to do. Of course, you could just leave the gaps (even if the tile is installed perfectly there will be gaps, especially with natural tile), but we never did.
Done properly, if tile was added where there was none before, either the baseboard should be replaced with it; or the tile should be installed in such a way that it is not cemented to the baseboard, then quarter-round installed afterwards.
Best Answer
Perfectly fine to drill through a baseboard.
I would drill a pilot hole from the side that still has the shorter-baseboard. That gets you lined up without worrying about splitting the new baseboard or pushing it away from the wall. Then drill the full-size hole from the taller-baseboard side, centered on the pilot hole.