Jared, choosing a personal trainer for your wife is bigger than the personal trainer having the right certifications.
In searching for a personal trainer for your wife, you will will want to choose someone who understands the science of exercise and knows how to apply it to help your wife achieve her post-pregnancy fitness goals. They should also be a great listener and know how to assist your wife in meeting her realistic fitness goals that your wife sets for herself.
The personal trainer should know the established pregnancy and post-pregnancy guidelines as it relates to putting together a program for your wife and should be willing to share it with you and your wife if you make request. They should also be aware of current guidelines relating to writing a prescription for exercise.
If you wife has any health conditions, the personal trainer should collaborate with her doctor to make sure that the exercise programming is safe for your wife. Even more so the personal trainer should have clinical experience or preferably an advanced personal training credential like the ACE AHFS, ACSM HFS, ACSM CES, or the ACSM RCEP.
It’s a good thing that you posted your question on this forum as there are many qualified people on this forum who can give you good insight on choosing a great personal trainer for your wife.
I wish you all the best in your search and congratulations on the birth of your second child. 8))