Thanks to everyone for the replies.
I'm on the latest version of PB, also in development so all of the PFODBxxx.Ini files are there.
MySQL does provide a method for retrieving the latest insert id, SELECT LAST_INSERT_ID(). Unfortunately, it has to appear immediately after the INSERT statement semicolon and PB doesn't allow that.
The other options seem difficult. Selecting Max identity is dicey, someone else can insert and in the meantime. Locking the table first would work but not sure if this would lead to a performance issue.
What surprises me is that this is an age old issue going back over a decade or more. I don't understand why others don't run into this problem (or why Appeon doesn't address it - seems like it was at one time by using the Identity Column on the Update Properties menu).
I guess it throws me back into the surrogate key debate. I understand there is passion both ways, to use or not to use.
I think I will redesign the table structure, eliminate the auto increment key column, and use a CHAR variable containing the name for the key, relying on the database to replace (CASCADE) any changes made to the key to the other tables referenced if required.
Once again, still perplexed as to why these issues remain after all these years.
And again, thanks to everyone for their input - invaluable.