Convert the process material to a shader material. Modify the following:
1)
uniform float initial_linear_velocity;
uniform float emission_angle; // <<<<<<<<< Add this
uniform float initial_angle;
2) Inside the vertex function, somewhere below if (RESTART || restart) {
, change the assignment of CUSTOM.x from base_angle * degree_to_rad
to (base_angle + emission_angle) * degree_to_rad
3) In the else bracket following the previous if, find where it says CUSTOM.x = base_angle * degree_to_rad;
and change it to CUSTOM.x += base_angle * degree_to_rad;
After the changes are done, you can send the shader an angle parameter by calling $Particles2D.process_material.set_shader_param("emission_angle", your_angle_in_degrees)
I hope this helps :)