DMCA takedown notices are for copyright infringement only, not for trademark infringement. The copyright infringement issues would be resolved if they removed the images.
However, you may well have the ability to get the domain name transferred to you using the UDRP procedure. Check the link below for more information. Unfortunately, a UDRP Complaint costs significantly more than a DMCA takedown notice, but in my experience they are usually very effective and fast and much cheaper than litigation.
By way of explanation, DMCA is not a hammer for the copyright agrioeved. Instyead it is a carrot to offer the web host. ‘If you take down the offending page I will not be able to sue you for contributory infringement, which I otherwise would.’
I see no reason the same offer could not be made to the host of a web page that contains an alleged trademark infringement—although the procedure is not specified by statute. The Lanham Act (trademark law) absolves the “innocent” publisher of a TM infringement, and I assume a court would extend the innocence to a web host. But as soon as you give a notice similar to a DMCA notice, they are no longer innocent.
Discuss both of these with your I.P. attorney, along with anything else that concerns you.
Information here is general, does not create a lawyer-client relationship, nor a substitute for consulting with an experienced attorney on the specifics of your situation.
One aspect that would require further facts is the use of your trademark in the URL. Your competitor has the right to start a domain called www.xyzcompanysucks.com. The UDRP Procedure is limited to only the clearest cases of cybersquatting, and would probably not work for you in this instance. You need to discuss this with an attorney in a confidential setting to fully understand your rights before large amounts of money are spent.