The burner pump seem to be able to take any kind of fuel, except the "processed fuel" that AAI Industry adds to the game.
I'm guessing its' a bug?

I asked "Fuel category compatibility by Glendening" to make a fix for this, but seems like it's supposed to support the processed fuel already and not possible to fix on his end.

https://mods.factorio.com/mod/fuel-category-compatibility/discussion/669a6df1d825ecfef0b3461f

Glendening answer:
Looking into it, that might be a bug with PUMPS.
It shows that it has the AAI fuel category enabled and attached to the burner pump.
But you can't insert the fuel.
There is nothing my mod can do since I can't add something that is already there.
You may want to send a bug report to the PUMPS mod.

Came to also post this.
The burner pump from PUMPS (offshore-pump-0) shows that it has fuel_categories populated with "processed_fuel" in the list. But the AAI processed fuel doesn't show up when you mouse over the fuel slot on the pump. Insert X-files music here

I am currently in the process of rewriting my library, bugfixes will then come, hopefully before the expansion drops

Sweet! Take your time.
Personally busy with Satisfactory atm, and will probably be going vanilla for a long time after Space Age drops. So excited!! :D