Check out the Wakefield track too - little more accessible than Billerica and a good bit easier for the little guys to ride.
Try hunting down a used bike - you'll save a bunch of $$. If you're interested in a micro, try inquiring if anybody has one up for sale through the local tracks' facebook pages - might have some decent luck that way.
If you're okay with a coaster brake, I have a 16" Redline Pitboss kicking around that'll might work for you. Been through a couple kids, so it's not pristine, but it does have flames.
Started out as shown in the picture, but has been through a number of 'mods' and riding seasons. Still got some good times left in it though.
Slapheadmofo Leisure Team
Sunday River Bike Park