I am not sure where you learned that you cannot or could not register a trademark for your group. Without knowing more about the facts, I will refrain from making any other comments on that issue except to note that there are many non-profit organizations that have registered trademarks, including non-profit professional, political, and fraternal groups. An organization can offer goods and services to its members, to non-members, to the community at large, etc.
As for dealing with the possible infringement, the issue to consider include whether there is a likelihood of confusion as to the source of goods or services associated with the marks—will someone think that other party’s goods/services come from your group and/or vice versa. Factors to consider, among others, are similarity of the marks, similarity of channels of trade or offering of services, and at common law you should consider geographic region, and, of course, who started using the mark first and in what manner.
I strongly suggest you discuss this matter with an attorney, sharing the facts, the manner in which you have used the mark, and the different communications you have had with the other party.
Please note that this information is for informational purposes only and is not legal advice and should not be construed as legal advice. Nothing herein is meant to create an attorney-client relationship.
You should hire an IP lawyer to sort out your business assets and trademark rights, since so far you haven't searched, filed or protected things properly.
Disclaimer: Please note that this answer does not constitute legal advice, and should not be relied on, since each state has different laws, each situation is fact specific, and it is impossible to evaluate a legal problem without a comprehensive consultation and review of all the facts and documents at issue. This answer does not create an attorney-client relationship.
You are mixing concepts. Not for profit is a creature of the IRS and has nothing to do with commerce, only taxes. Your club is a business, and can apply for a trademark.
A trademark only cover a specific class, so unless you do music business, you probably cannot register in that class and the music group can. Your club and the music group can have the same trademark and be in their respective classes. Think Apple computers and Apple records.