I haven't camped there but there are some pretty easy trails that your little man can handle, and there is a lake in the park that you can rent canoes and small boats to go putting around in. They do allowing fishing there as well. I am not positive but I believe they have a pool there as well.

Would this Be Harrisburg NC?

You should look into McDowell Nature Preserve down on Lake Wylie its right of HWY 49 aka South Tryon and has everything you listed out.

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Funny you ask about this.  I am currently planning a first car-camping trip with my 4 year old.  LNSP seems to be our leading choice right now, especially for the proximity to home if we have to bail.  I think it has everything you are looking for.  I am planning to take him (and his bike, of course!) on a weeknight so it won't be as crowded.  If one night goes smoothly, we may stay for two... playing it all by ear with contingency plans in place!

Was considering LNSP, but it's actually further from me (according to google).  Might still give it a go.   

Just posting a follow-up..

Took my 4 year old son to LNSP campground last week, Tues-Thurs.  We had a blast.  Some sites are a little close together, so scoping it out before making site reservations would be good.  I could see it being pretty crowded on the weekend.  On the flip side, it would be a good place for a couple of families to camp side-by-side, especially where we were... (Site 14 -- very close to Site 13).

We hiked, did pedal boats, threw rocks in the lake (his new favorite thing to do...).  He loved cooking over the campfire.

The host and ranger were both very nice and accommodating.  They warned us about the wildlife, and they were not kidding!  The squirrels are very bold and will take your stuff even while you are sitting right there.  Just because you washed your camp utensils/pots/pans, don't think they won't get up on the table and get IN your stuff and lick anything that might have had food on or in it at some point.  They will steal your food, too.  I also came face-to-face with a raccoon one night as he was wandering over to our neighbor's site to eat the dog food they left out.

They sell firewood for $5 a bundle which is a little pricey, but each bundle was enough for an evening's fire, and worth it to not have the hassle of bringing our own in the car.

The host told me there were some other Trailblazers there on Tuesday night, and I meant to walk over and say "HI", but I missed them.  So, if you are reading this, ...."HI!"  (not sure who it was, or if they are on the forums).

ETA:  Since the canopy is very thick there, all the sites were very shaded, so even though it was in the 90s it didn't really feel that sweltering.