Will it be accurate when switching to synthetic oil? No, but the error is on the side of good.

The system cannot determine the grade or type of oil, this information is programmed into the algorithms of the program. So, the program is set up for the use of oil that meets or exceeds the oil specified for the H3.
The spec for the H3 is GM6094M, and is not as stringent as GM4718M, the spec for syn ethic oil. So, if you use a better spec'ed oil, the indicator will not be accurate, but if it says you have 30% left, it will more likely be 40% (this is not an exact number).
So, there is no problem using any oil that meets and exceeds the GM spec required for the engine in the H3. For example, Mobil 1 states it meets 4718M, which is a higher spec than 6094M. In fact, the owner's manual for the H3 states that under certain conditions (< -20 degrees F), the use of a synthetic oil is recommended.
The big problem would be using oil that does not meet or exceed the spec, and then relying on the monitoring system.
Example would be on a Vette, which is programmed for synthetic oil, and someone decides to save money and put in the oil spec required for a H3.
Since synthetic oil degrades slower than regular oil, the oil life monitor system on a Vette may display 30%, but the actual oil life might be 20%.