Unfortunately no. No options to change the behavior of the build-in screenshot tool are exposed.
The new screenshot tool in Ubuntu 22.04 and up is integral part of Gnome Shell. No options to change the behavior are exposed. Thus, it would require editing source code to change the behavior.
The only way to have the screenshot functionality work in a different way is to use an external tool that exposes the configuration you want. You can replace the existing keybindings with commands that call your tool with the desired options.
The classical gnome-screenshot
tool remains available, and continues to work fine, also under Wayland. It allows to specify where to save screenshots and under what name.