Clutches don't care if the oil is synthetic or not. Synthetics generally have a high temperature withstand before breakdown. That's why I use a synthetic blend oil, in case the engine gets hot and there is no airflow over the engine cooling fins. Just a bit of extra protection.
What clutches do care about is the additive package blended into it. Some additives will make the clutch slip, such as what is found in today's Auto oil offerings.
That's why you want a motorcycle specific (for wet clutches) oil. Or experiment with the oil intended for diesel motors.
I suggest you use a motorcycle specific oil until you get the clutch sorted. Then experiment with other oils if you think that is a good use of your time.
In fact, for the period of diagnosing clutch issues, why not go for what Honda recommends for the bike, even Honda Branded. However, I would certainly avoid oil with added Moly, while you get the clutch sorted.
Anyway you can chalk up the added cost of Honda oil to a diagnostic aid/tool. And if it fixes your clutch issue, it's actually cheaper than replacing a clutch.
Back in 1975, I acquired a 1974 Honda CB550, that was wrecked. Had 1500 miles on it. I replaced all the bent bits and got it running pretty well. But, the clutch would slip and it could not be adjusted out. Speaking with the limping crash survivor, he had added STP to the engine, thinking it was a good thing. OK for the engine, NOT ok for the clutch. It took ten oil changes for the clutch to regain close to full operation and not slip when hammering the throttle in top gears. I don't think it ever fully recovered. But, I wasn't too disappointed about never being able to pop wheelies or do burn outs.
Cheers,