Hi - you should definitely sign him up for the gifted and classical tests this year. A few spots open in each class each year as kids leave the city etc. Also, around 3rd grade, the class sizes can increase to around 31 kids so some additional spots open up.
YOu can also see what happens with the magnet and open enrollment lottery.

There is probably a school that could challenge him more. If you're open to moving during this school year, you could call around to some of the neigbhrohood schools with good reputations to see about a transfer.

My understanding about Catholic schools is that religion is rather integrated throughout everything (well, maybe not math, but it's not like there is 45 min of religion a few times a week. You have to be fairly open to ongoing religious references. But my understanding is also that plenty of non-Catholic kids attend them. Probably more of a stretch if your family isn't christian or religious at all. But they can certainly be a great education value compared to other privates.