Finish cutting board with shellac on bottom and oil on top

What issues could I run into with this?

Mainly, you'll run into a not-very-durable finish on the sides and bottom, and possibly a weird transition between the oiled part and the finished part. Water and other liquids will likely seep under the shellac film where the sides meet the top. Water will no doubt contact the sides and bottom of the board and cause the shellac to cloud. Mechanical damage from knives and other kitchen implements will scratch or chip the shellac.

I did a fair amount of searching but can't find any references to anybody else doing something like this. i.e. a seal on sides/bottom and a non-sealing oil on top.

That should be a strong hint that it's not a plan worth pursuing.

I'd guess you're thinking of shellac because it's food safe. But although shellac is safe to consume, it's not a good finish for standing up to the kind of abuse that cutting boards get.

A much better plan would be to finish the whole board with food grade mineral oil, butcher block oil, etc. You'll need to re-apply it from time to time as the oil soaks in or is washed away, but it's not something you need to do often. Even with your hybrid oil/shellac proposal, you'd still need to periodically oil the top; using oil on the entire project won't increase that time much at all since the bottom and sides likely won't need more than a swipe or two with the same rag.