Caius, Caius. What is the exact reason for the name change? As said before it should be made easier or more clear (but that is arguable). From Caius Flaminius to Caius would be the most logical and effective doing.

I will give you another reason for making it into Caius. Observe the language beauty when you regard the three vowel-ish letters in-between whereof one sounds not as a vowel if I'm correct.

Now in general:
Let us compare the two names:
* Caius
* Garcilaso de la Vega el Inca

The name Caius is clear, short, effective, and arguably a language beauty (to me), while the other is unclear, long, and ineffective as it possesses juxtaposition of too many elements.

I know I know: it is difficult to counter these arguments, or those already made before me. But as you now notice, the only only only only effective name is Caius. I vote it to be