Reflection probes will help, and may give you enough of the effect you want.
Your dynamic object is lit by a combination of multiple light contributions:
- a base diffuse contribution, which comes by default from the Global Diffuse Map you set in the shading environment, but may be overridden by baked light.
- a specular contribution, which comes from the Global Specular Map in the shading environment, but may be overridden by local reflection probes and by screen-space reflections.
- dynamic contributions from direct lights.
So say you want the color of the surrounding room to bleed onto the character when your character walks from a yellow room into a red room. If you place reflection probes in those rooms and bake those results, then they will feed into the specular contribution for your character. This will produce the color bleeding effect in proportion to how much specular contribution gets blended into your character's materials. So you'll see more of the color bleeding at sharper angles and on materials with lower roughness.
But the diffuse lighting on the character will not be affected. So if you look straight on at a surface, or if your surface is super metallic, you won't see any color bleed at all.
You can also of course add in some low-brightness omni or spot lights into the scene to fake the effect, but obviously that's not the ideal way to go.