you don't need a visa for CPL as they don't class it as significantly improving your flying skills
Well, we disagree there then, That would be the day....
If CPL training does not increase your skills, you've been to the wrong place.
A lot of people choose to do the Initial Commercial on a twin, which requires a visa.
Tell Immigration upon entry that you are here for recreational flying as not everybody is familiar with the term hourbuilding or timebuilding even though it is somewhat self explanatory.
Recreational flying does not imply any training.