Your next step (which should be your first step) is to figure out if "your phone" uses late edition of PowerDelivery and can negotiate anything above 5 V level.
If it supports PD and can use the elevated 12-V voltage profile, you can proceed with:
(2) Make a 12V-to-5V switcher, to provide the 5V source for default power path (because everything must start with 5V)
(3) Since you already made your selection for BM92A50MWV chip, you need to read carefully its datasheet, not just the matketing description of it.
(4) Then you will need to select any MCU that supports I2C interface, and write a program which will communicate to the BM92A50MWV chip which power profile you can/will offer, which would be obviously a 5-V (with whatever current you can manage with 12-to-5V switcher), and the 12-V profile with whatever current your battery can deliver. This will allow the BM92A50MWV chip to spill out its right capabilities to external cable and to the connected "your phone".
(5) upon successful completion of PD exchange protocol (done by BM92A50MWV ) and accepting the phone request for the 12-V profile, the chip will likely automatically switch from 5V to 12V path, although it is not clear from the datasheet.
(6) before doing all this, it is highly advised to procure a reference platform for the BM92A50MWV if you can find one, with code samples and internal register definitions. If the design doesn't exist, even after contacting ROHM engineering, change your selection of CC controller IC, and start over.
That't about it.
There is no such power splitter on the market that I have seen, and if it did, would violate the USB spec and defeat the purpose of Type C PD spec.
But technically, you could do this on your own but may not be worth it. You could just tap the VBUS and GND lines of the power adapter and tap power from that.
You will not get any voltage unless you plug the adapter into the laptop. And because the laptop is the device in control, it could ask for 5V, 9V, or 12v (or even 20V if a 20V charger was plugged in), so you might not get the voltage you were expecting. Also, in the standard, the device knows the capabilities of the power adapter, but your additional load throws that off and the total load can overload your adapter.
USB Type-C has complicated power negotiation. If the laptop or other device is not plugged in, it will turn off its output for safety. The default voltage is 5 Volts. Higher voltages require special signalling between the device and power brick. And the laptop could tell the brick it no longer needed power and voltage would fall to 0.
Anyway, that should give you some idea on the ramifications of what you are asking for. Here's some technical info on it:
https://www.ti.com/lit/pdf/slyy109
Best Answer
The USB Type-C doesn't work as you described. The load of ideas how the port should opreate (as dual-role port) is described in the original USB Type-C specifications, which are freely available from USB.org, as part of entire USB 3.2 zipped package.
Initially none of Type-C port will have any voltage on VBUS, so you can't use it as detector who is provider, and who is consumer. The role of a port is determined by pulls (up or down) on CC pins. And the CC wire goes from one port to linked port over C-C cable. Once the right combination is established and recognized, the source will put out VBUS, and the consumer will take it. For details please examine Section 4 of Type-C specifications, all variants of connectivity are pictured there.