Having re-read the story, I'm convinced that you're simply misremembering Spider Robinson's "Melancholy Elephants."
The conflict is over a bill intended to grant perpetual copyright:
"You know the system we have now, unchanged since the mid twentieth-century. Copyright ceases to exist fifty years after the death of the copyright holder. But the size of the human race has increased drastically since the 1900s—and so has the average human lifespan. Most people in developed nations now expect to live to be a hundred and twenty; you yourself are considerably older. And so, naturally, S. '896
now seeks to extend copyright into perpetuity."
But every existing song is already tracked and protected, with the result is it has become increasingly hard to find new songs to compose:
"Did you know that at present two out of every five copyright submissions to the Music Division are rejected on the first computer search?"
The old man's face had stopped registering surprise, other than for histrionic purposes, more than a century before; nonetheless, she knew she had rocked him. "No, I did not."
"Why would you know? Who would talk about it? But it is a fact nonetheless. Another fact is that, when the increase in number of working composers is taken into account, the rare of submissions to the Copyright Office is decreasing significantly. There are more composers than ever, but their individual productivity is declining.
The detail of the composer who is crushed to find out that the tune he wrote is one from years earlier is straight out of the story:
"My husband wrote a song for me, on the occasion of our fortieth wedding anniversary. It was our love in music, unique and special and intimate, the most beautiful melody I ever heard in my life. It made him so happy to have written it. Of his last ten compositions he had burned five for being derivative, and the others had all failed of copyright clearance. But this was fresh, special—he joked that my love for him had inspired him. The next day he submitted it for clearance, and learned that it had been a popular air during his early childhood, and had already been unsuccessfully submitted fourteen times since its original registration. A week later he burned all his manuscripts and working tapes and killed himself."
The databases aren't burned immediately, but for the time being perpetual copyright is avoided, and there's hope that maybe that forgetting will be permitted:
"If you live long enough," the senator said slowly at last, "there is nothing new under the sun." He shifted in his great chair. "If you're lucky, you die sooner than that. I haven't heard a new dirty joke in fifty years." He seemed to sit up straight in his chair. "I will kill S. 4217896."
It was collected in the Robinson collection Melancholy Elephants (1984) and the Asimov/Greenberg anthology The New Hugo Winners (1989).