Can I configure home battery power like a UPS - where it constantly self charges and auto transfers?
Yes you can, this is common in places where there are frequent power failures. When the grid is available it is used to charge or maintain batteries, and when it goes out the batteries are used to power the home.
A simplified overview of the system is as follows:

When the grid is available, the transfer switch will be set to power the loads directly from the grid, and the grid will also power a charger to keep the battery topped off:

When the grid goes down, the transfer switch shifts the loads to the inverter (quickly so there is no interruption), so now it works as follows:

When the grid recovers, the transfer switch will go back to its original state as explained beforehand, and the charger will kick in to recharge the battery.
However installing a system that provides backup power for your entire house is going to be very expensive as you need a big inverter and a huge battery bank to handle the load. It is more practical to have some essential circuits on backup power (such as lighting and some appliances), and to manage to live without those which are not essential during the brief outage (such as maybe air conditioners or water heaters).
The overview above is rather simplified, and making such a system yourself is IMO not going to work out well by just buying a transfer switch and a separate inverter and charger. What you may look into buying is something called an inverter \ charger. Magnum, Schneider, Victron, Voltronic etc. make them.
Edit:  I noticed that you mention  Blue Planet Energy, I am not familiar with them but maybe it is not just a battery and has required components already built in like a Tesla Powerwall, you should ask them about it.