A grill requires massive ventilation. Rethink that. Your are in the land of 1200CFM blowers and Ventahood. With an well engineered Make Up Air system 6-10K for the package depending if you need heated MUS.

I don’t have a grill... but I do have 6 burners. I did a lot of research and narrowed it down to the Italian brands & Ventahood. My ultimate choice was a Zephr which offered more power and LED lighting. I LOVE MY ZEPHR.

If you are venting an island the units are much more expensive than venting a stove that is adjacent to a wall. PLUS the installation takes a lot of time. (Ours took 2 pros a day — and they had the advantage of an open attic above.)

Any hood with lights can have LEDs to replace the other bulbs I did it a long time ago with my Vent-a-hood which BTW is my preferred venting system. An indoor grill is IMO always a bad idea and to contain the smoke and grease you will need a very high CFM hood which by its very nature is loud and every time the vent pipes need to bend your CFM needs to rise . Not sure why the stove can’t be vented directly out but without seeing the kitchen and layout hard to comment on that.This is a small space so even more important to get smells, smoke and grease out quickly.

i would take a look at these - depends on the width of the cooktop. Not cheap but they are solid and made for the Miele ranges

https://www.mieleusa.com/domestic/product-selection-of-range-hoods-2984.htm#5