You've got several problems at play here, some of the technical, some of to do with managing the expectations of your client.
Firstly, the technical: a 6 megapixel image will give you around 3000x2000 pixels to play with, which at 150dpi will result in image around 30 x 50cm. You want an image that is roughly 5x larger than that, so you either need to get a file 5x larger in size, or settle for an image that's going to reproduce around the 30dpi mark (which will look terrible). Pragmatically speaking, one thing that frequently gets overlooked is viewing distance. The further a print is going to be viewed, the less dpi required, so a 150 dpi image will look OK from across the room, but blurry up close. Large format prints rarely need to be viewed up close, which is why printers will often specify sub-300dpi images. The simple fact is there is an upper limit on the size of an image; it's simply not feasible to get a 300dpi image at 189 x 254cm. It would weigh in at 650 megapixels! Whenever asked how big an image should be I always answer as big as you can possibly get.
Now, a practical solution for you: get a sample printed using the supplied image. Tell the printer you want a sample showing the image scaled up to final size so you can check quality, and select a selection of the print that you think would be the most problematic (typically a section with lots of detail or noise). If the client baulks at the extra expense ask them if they will be happier to pay for a full banner that they might not be satisfied with. Then present that to the client, pin it on the wall that the final banner is going to hang and assess it from whatever the normal viewing distance may be in the environment. You might find the quality acceptable; for example, if the banner goes behind a reception desk the extra couple of metres viewing distance can make a soft image less noticeable. You may also be pleasantly surprised that the client doesn't notice any quality issues and signs off on the sample; I've found many clients simply don't notice the kinds of details that designers notice. Happy days.
If the quality isn't good enough you'll need to source a new image, but at least this way you've managed the client's expectations, kept them involved in the process and covered your ass.