Thoughts on a Greek —> English name?

My husband and I are both children of Greek immigrants, and most of our family is still there. We go back often, speak the language, etc. and it’s our intention to make sure our kids feel the same connection to their heritage (current plan is to have them spend summers there with grandparents and family!). So, it’s important we have names that work in both Greek and English (and Italian, because half my family is there…but I digress).

My husband is very into the name Βασίλειος (Vasileios), typically called Βασίλης (Vasilis), if we have a boy. In English, this would be Basil. I’m less sure, even though I like the name in Greek.

I have a two-part question: (1) what do you think of the name Basil, potentially nicknamed Baz? Is it too “weird” / does he risk being bullied? And (2) ideally, we’d want to spell it “correctly”, which would mean Vasileios or Basileios. In a pinch, Vasilis or Basilis would do. How much worse is that than Basil? Should we just name him “correctly” in his Greek ID (he’ll be a citizen through us), and call him “Basil” in his US birth certificate?