If BATS has the best price or matches the best price available in the market, your order will execute at BATS. If there is a better price available at another market center displaying a protected quote, BATS has to route your order to that exchange.
Continuing with the example, lets say I'm trying to hit my price, and only 300 shares are available on BATS and another 700 are available on some other venue. Would BATS send an order for the remaining 700 to the other venue, or does it only route the order and not chop it up?
Depends on the routing instructions in the order, but generally speaking if BATS matches the best price then BATS will match as much as it can against the liquidity that is available at BATS and route the remainder. See page 2 of this PDF for BATS: https://www.batstrading.com/resources/features/bats_exchange...
