I think you need to be really careful about this. Even if you're both talking regularly and openly about what your new relationship is and what it isn't, there's a lot of potential for one of you to get out of sync with the other and one or both of you could wind up hurt.
When you decided to divorce, I hope part of that was due consideration of why you got married in the first place. Now, it's time to give due consideration to why you divorced and let that shape the new relationship. I wouldn't count on either of you being an "all new person"
And specifically, unless you have in mind to get back together with him, I think a nice, non-romantic birthday card with a friendly but non-romantic personal note would be more appropriate than flowers.