Do I need to freeze this puree and syrup?
It has been said:


  Better safe than sorry. You haven't measured the absolute sugar or acidity levels. Freezing it will do no harm. –  SAJ14SAJ Jan 29 at 1:47


I'm definitely not an expert with baking, but I've worked with fruits in other contexts. The only time I would be worried about freezing would be if I added oil, like for a marinade or such. In your context I think freezing is fine, and sounds like it would be a convenient way to make larger batches to reuse throughout the year.

PS: That puree sounds awesome, I might have to try that to marinate some chicken or pork chops!