Currently, when I develop my dApp, I select a BP to push my transaction to blockchain. But I am wondering, what's happen when thats BP to become a standby BP?
Any working nodeos endpoint will enable you to push a transaction to the blockchain. It doesn't matter if it is run by a BP in the Top 20 or not.
You can use the answer here, https://eosio.stackexchange.com/a/1221/3333, to get a list of nodeos endpoints that you could use. Just use whichever endpoint you find reliable.