No. The danger that is seen in this situation is, of course, the possibility of a counterclaim for legal malpractice. But sometimes that danger is very slight.
I have experience on both sides of legal malpractice cases. When I have been asked by firms to consider suing a former client for an unpaid fee, the first thing that must be done is to evaluate the possible merit of a counterclaim for legal malpractice. Sometimes it is clear that the potential counterclaim would be without merit, and in those cases, if the firm's former client is solvent, it makes sense to go forward. In one I had recently, the former client had been threatening a legal malpractice action before I even got involved in response to correspondence asking for payment of the arrearage. I determined that the potential legal malpractice case was most probably lacking in merit. We went forward with the suit for fees. Once the former client had counsel, that counsel also starting make vague threats about a counterclaim. I called his bluff and explained that it would be frivolous, and he would be creating potential exposure under the sanctions rules and Chapters 9 and 10 of the Texas Civil Practice & Remedies Code for his client, and maybe even for himself.
It turned out that the malpractice threat was just a bluff being used for leverage, and the case settled on very favorable terms for my law firm client.
So, I would advise not to automatically walk away from those unpaid fees.