The only true non-toxic finishes that I know of are Shellac and Tung oil (pure Tung oil not thinned with mineral spirits). The problem with Shellac is getting quality. Most brands, like Zinsser, use low quality flakes and add Methyl Isobutyl Ketone to the alcohol solvent. The only way to get true Shellac is to buy your own flakes and cut it yourself.
With that said, if you want to buy a ready made Shellac then Zinsser Bulls Eye is a good way to go. It's readily available.
Tung oil can have the same problem. Buy only 100% pure without any additives or distillates.
Both of these finishes are great for any interior woodwork including cabinets.
I wouldn't be overly concerned with the polycrylic you have. It is easy to work with, dries fast and should pose no major health risks once dry. Plus you already have it.