Look at your flat iron and see if its rating is less than 1875w. Then
you should be OK.
OK? How. You mean will it literally work? Maybe. Is it legal? Definitely not.
You probably won't go through customs anyway, and if you did, they probably wouldn't confiscate it, but the converter you bought definitely is not in compliance with European electrical codes. See how it connects to the power sources with two round pins. That kind of a plug is known as a "Europlug" because it fits in receptacles all over continental Europe.. Technically, it is a CEE 7/16 or type C plug. This plug is limited by code to 2½ amps, or 575W at 230V.
Also, the 125V plug on the CHI device is probably "Polarized" (one blade wider than the other). This polarized plug is demanded by UL to meet certain safety standards, and is only effective on polarized power system. Italian power is not polarized, so these safety considerations would be defeated. (Even if Italian power were polarized, the fact the the Europlug can be plugged in either way would defeat the polarization 50% of the time.) If you have to take it with you, and I'm not recommending you do, never leave it plugged in unattended.
CHI might prohibit use of it's appliances with a converter because they know the safety standard won't be effective if used with a converter. They just want to cover their rear ends if something happens, and they get sued. (We told you not to do it.)