You can't bake directly with Filmic but you can composite the texture baked.
Baking with Filmic
Warning: before baking check if:
you must first create an image using 32 bit Float, this will allow to save values higher than 1 (this is the most important part, if you don't create the image correctly you will need to rebake everything)
assign the texture for every shaders used by the baked mesh, make sure it's the active image by selecting it if you have multiple textures
bake the lighting
import the image in the compositor and plug it to the Composite node
change your render resolution to match your image, don't forget to set the percentage to 100 %
in the Scene tab in the property panel, change the Color Management setting to Filmic instead of Default and you can eventually adjust the contrast in the Look setting by default it's set to None (does the same as Base Contrast)
do a render (F12) and save the new image, I recommend you to use TGA
You can then update your lighting, bake it and press F12 to have your new lightmap.
Do it for every bakes including the skybox to keep a consistent lighting and avoid "burned" areas.
Using it in Godot Game Engine
To use it in Godot, export your OBJ and uncheck "Write Material" this will allow to have only one material slot. Export your lightmap as TGA, this format is faster to load than PNG and use RGB instead of RGBA to save some memory.
In Godot select your texture, in the Import tab set the Mode to Loseless and uncheck Filter (it blurs the texture). Add your .obj into the scene, create your Material, assign the lightmap as Albedo and in Flags check "Unshaded".
enter image description here
While playing if the skybox (the sphere with the sky texture) isn't visible increase the Far setting of the camera. You can set it to 1000 instead of 100.
How to use it correctly
When using baked textures the size of your game can become heavy, that is why I recommend you to have small and few rooms, like for example a 3D puzzle game like Portal.
If objects have to move or be collected the lighting won't be updated. That is why I suggest you to make them transparent, like holograms, so they fit in the game, like in this fully baked game: https://youtu.be/H-E7_mn73BI
What is the benefit of this technique
Having fully baked lighting allow to have a great looking scene that uses almost no ressources to run, it can run without a graphic card at 60 FPS even on a web browser or a mobile device as long as you don't fill the memory.
Later denoising will be added for Cycles baking allowing to get a better result with less samples.