I doubt that the engine would notice a difference in which oil you use.
If your engine is carbureted however, what will make difference is your jetting. Going to premix isn't simply removing the pump and mixing up a batch of fuel.
I have lots of experience with vintage two cycle motorcycles and know that when one takes off the pump, carb jetting must be reviewed. I haven't switched from the pump on my machines because they're very reliable and convenient. However, I have read accounts of those who've done it and they advise that re-jetting is required.