"When I look at this it seems to me that the 12' span between the beams is too long."

Well, duh!

"Lumber supplier said it should be fine."

Yeah, for a roof. I'm sure this guy has plenty of experience.

"My question is should I run a 3rd beam down the middle or am I ok as is? I also was thinking of doubling the 2 end joists for more support."

Now you're on the right track. 2x8 PT should not span more than about 7'. Double up all the rim joists and run a double joist wherever the decking will butt end to end. I'm not experienced with the composite Azek, but for any wood deck 16" joist spacing well exceeds the load requirements.

Azek has the least span strength of all the manmand. The stuff requires at least 12'' on center joists with any kind of angle install its more like 9 / 10 '' on center. J.