Get someone to put the same asphalt shingles that are on the roof onto the overhang for the window. Paint and tape aren't going to cut it.

It might be cheaper to add acoustic insulation from the inside and then add new drywall over the intrusion.

I'd price it out with a roofer. It may not be very expensive.

is there a gutter on the roof above it? its when the rain runs down a larger roof surface and lands on the metal that the noise is really bad.

a gutter may be the least expensive solution then, if a roof above is draining all of its drops as well as whats actually hitting it.

friends have a metal roof on their home, they have sound insulation above their ceiling, but still in a cloud burst its hard to have a conversation.

I love the sound of rain. Are you sure you couldn't learn to find this peaceful and soothing rather than irritating? Would be a lot cheaper in time and $$. :)

I find that interesting that 3 roofing companies would NOT know how to roof a bump out. In essence, they would REMOVE the metal, lay a proper roof deck (plywood + tar paper) and then add the identical (or nearest approximation) shingle to the one on your house. Done.

The problems I see would be the "white" shingle being a problem (I don't know anyone that carries a white option), the "size" of the project might not be worth it or the budget might not be appropriate.

This is more than doable from a roofers point of view...but you MUST remove the metal and complete the decking PROPERLY and then use a shingle that is NOT white. Somewhere in your "requirements" might be the reason that 3 different companies would not know how to do a simple job as this.

Love sleeping under a tin roof when it rains! Not for everyone though.

It should be a straightforward job, but one thing you might want to try in advance is painting on a white elastomer paint. You'll probably need to go to a Sherwin Williams commercial supply and I think the smallest quantity comes in a 5-gallon bucket for about $180, but that's still cheaper than a roof job if you only use a cupful. You can have the elastomer tinted if preferred. It's used on commercial metal roofs like aircraft hangars. You might find an organization like a church where you can donate the remainder of the paint after the job is done. Another option might also be to remove the drywall ceiling of the bay window between the rafters and use spray foam for insulation and then replace the drywall.

We have a tin roof over our back patio and it is very loud. I personally don't mind the sound, but if it gets under your skin it would be hard to ignore. If you can afford to address it properly, maybe it will look nicer with shingles on it anyways. I would probably try wearing ear plugs at night, when it's supposed to rain, until you can have it fixed.