Classification accuracy based on top 3 most likely classifications
This is called top-3 accuracy.  See https://stats.stackexchange.com/q/95391/2921, https://stackoverflow.com/q/37668902/781723, https://stats.stackexchange.com/q/156471/2921.  Sure, you can use it.  Whether top-1 accuracy or top-3 accuracy is better correlated with user satisfaction will depend on your particular application and on your users.