No. You are hired based on the certificate. Shuttle and Republic have some different procedures and you can not jump back and forth between them. You're either on shuttle, or republic.

The only way you can change certificates is if you are going for an increase in pay. IE: FO at republic and you can hold captain at shuttle, you can upgrade to shuttle. However, you can not do that in the right seat because the payscale is the same for all types.