How do I tell it that it's allowed to take more current?
To start, there are Type-C basic "fail-safe" specifications, which use CC wires and indicate with various pull-up values that your supply has 900mA, 1500mA, or 3000mA capability all at +5V. Then the phone can take the current per the CC connection level, see https://electronics.stackexchange.com/a/382116/117785
If you need more (5A @5V or higher voltage profile), you need to implement Power Delivery specifications.
How do I implement USB-PD?
To implement USB-PD, you have three options:
Read, understand and implement a subset of PD specifications. Keep in mind that the latest PD specs have 870 pages of text, while USB3 specifications have only 550 pages (and USB2 only 650 pages). You will need an IC that starts communicating with CC lines (communication channels of Type-C specification first), and then switches to PD physical layer protocol (BMC). Then you will need a 32-bit-grade MCU to implement PD protocol itself and its policy engines, including discovery of cable properties/capabilities and link partner capabilities. This will take maybe just few thousands lines of C++ code and several engineering man-years to implement all this.
Take a Texas Instruments PD chip or Infineon/Cypress PD chip, and ask them to implement a specific set of power profiles that your power source is designed for. But they likely won't talk to you unless you guarantee to buy their chips in quantity (say, 100,000 per week). And they won't share their firmware code with you.
Go to AliExpress/eBay and get a little board that implements most of the necessary controls, search for "power delivery trigger board".
The choice is all yours.