I'd put put a bosh 5 prong relay on the output side of that button.
Wire the illuminated button as required but instead of + output from switch to accsy:
On the relay
85- Ground (-) you can loop it together to 87
86- directly to + output on that switch diagram.
87- Directly to good ground.
30-current ground to fan assy.
What will happen: When you flip that switch on, 86 will see (+) fron your new swtich, 85 will already be (-), causing pins 87 and 30 to connect- grounding your fans and turning em on. Just safer that way.
I would do it that way instead of trying to run a lot of current thru a small switch like that. See type B relay on the bottom, that's what I am referencing below
