By opening your "project" file (you share with us), you use Color Ramp node set Pink(left side) > Black(right side) that works just fine for me.
On other hand in your screen the Color Ramp node is set Black-Pink ... this setup generates black in all bake.
The problem lies in your Layer Weight node. This node can't be handled by bake as intended, because the factor is considering face normal direction (angle) pointed to camera view. For baker all faces are calculated as front view. When ColorRamp is set from Pink to Black. First color(Pink) is taken as color assigned to normal pointed to camera (front). So if you want bake pink emission, keep Pink at Left.
Just to be clear, you can't bake the gradient (pink centre with black border) since by observing a scene you'r changing dynamically point of view, it means what is black border fro one point of view are not the same black faces from another point of view.
You can set node-tree with pink at centre of object and black border to be baked, but it would be correctly seen only from specific point of view, and that is not what you are looking for.
I don't have experience with Three.js but it use WebGL so probably it works like Sketchfab ... in this example ground and tree trunks are baked (you already know how to).Tree leaves (spheres) are just simple Principled BSDF node with Base Color Pink, Metallic 1, Roughness 0.05, Emission Pink
Sketchfab is able to import and translate into its ecosystem. Textures should be packed and use naming convensions to be assign automatically (but can be uploaded and assign also manually. Bloom is set on Sketchfab site ...
The light difference between Cycles and Eevee is that - Eevee doesn't support mesh light (directly), so in Eevee is your scene light just by light objects, but Cycles render is light by emissive tree material plus by light objects ... that is why cycles looks lit by light more than in Eevee.
Note: Some of your tree normals are orientated inside ... it can generate some issues. To fix the orientation, select your objects, go to edit mode, and press
N to Recalculate Normals.