The EVE Online bots I have seen so far don’t support running in the background. You can use an OS-level containment to do that. Besides the classic virtual machine, another option is RDP Wrapper.
In that case you will use two instances of the BotLab client, and select a game client with each one.
No need to purchase separate credits. You can run any number of bots using the same account, and the same key.
Of course you will still see a separate entry for each play session in the Reactor.
You can use the mining-site-location setting in the bot settings to use bookmark names from your game client.