In the US, copyright damages are basically profits made times a multiplier based in willfullness. Once you have received a cease and desist letter, any further copyright infringement gets mega-multiplied damages.

Other areas of the law work similarly with big increases for willfulness.

One example is Hulk hogan vs the-media-company-that-will-not-be-named. Not only did they not remove the video after a cease and desist, they also refused a judge's orders. This led to epic damages in the final court ruling - 140 million and it put them out of business.

So this is something that cannot be lightly ignored. If you do anything other than stop doing whatever the other company is mad about, you absolutely need to check with a lawyer first.

This will cost money. I'm sorry.