I have no idea what a uflymike or a bose whatever is, but if you're talking about a headset that fails when batteries run dry, you'd better have a backup. If you lose all communication capability when the batteries run out, then you do have a problem.
It's not a TSO issue, it's an operational and safety issue.
The FAA defines the TSO under Part 21.601 as "A Technical Standard Order (referred to in this subpart as "TSO") is issued by the Administrator and is a minimum performance standard for specified articles (for the purpose of this subpart, articles means materials, parts, processes, or appliances) used on civil aircraft."
Products or applicances that do not meet TSO standards will not be issued TSO approval. Also, products manufactured outside the US, with a few exceptions, are not eligible for TSO approval.
For the purposes of certain operations, certain communication equipment is required. For example, under 14 CFR Part 135.165, two headsets are required, or one headset and one speaker. Operation with certain persons aboard may require the use of an additional headset. An example would be an inspector, who requires access to the flight deck, and a headset. See 135.75. Under Part 121.349, two headsets or one headset and a speaker is required for flight under IFR or in extended overwater operations.
91.511 sets the headset requirement in large aircraft in general. It does not specify a TSO.
125.317 provides a similiar requirement.
The requirement for headsets are found in numerous places, but none of them specifically cite a TSO for the type of heaset.
23.1431 requires that certain electrical equipment, when the pilot is wearing a heaset, be able to transmit to the headset an aural warning. If the headset fails because the batteries die, then the pilot may not receive the warning. That particular regulation can be a little confusing, because the requirement is that an aural warning can be heard while any type of headset is worn, not that the equipment be compatible with any kind of headset. In a nutshell, if the equipment makes a beep loud enough to be heard no matter what you've got wrapped around your ears, its' golden.
Advisory cirular AC 65-15A cites the headset or speaker as basic components of any communication system. Why introduce a component that's known to fail?
The headset is a pilot item, not an aircraft item. This is why you don't find prohibitions or requirments for specific types of headsets in aircraft flight manuals. AFM's and communication equipment data does not and will not cite specific headsets to use, or specific TSO's that must be used in the aircraft. However, at the same time, the reason for having a TSO is to establish a baseline standard for equipment.
If you were to have an accident or incident because you failed to copy an ammended clearance, failed to hear a warning or tone, or for any other communications failure due to your non-TSO'd headset (took it off to change batteries, perhaps), the liability would be all yours and the penalties inescapable. An attorney would have a field day, as no doubt would any field inspector.
If you are using a system that's picking up your speech through a CVR (cockpit voice recorder), in addition to the area microphone, your headset must be capable of delivering transmissions to the CVR. 14 CFR 23.1457 sets this requirement. It's a certification standard for the CVR, but if your headset can't comply with the certification standard, then you've made the equipment unairworthy, and are still in violation. 27.1457 and 29.1457 set the same requirements in rotorcraft.
Having said some of the above, merely because you don't find a regulation that says your headset or other equipment must be TSO'd doesn't mean the FAA doesn't expect it to be. The FAA doesn't issue production certificates for headsets the way a certificate is issued for aircraft appliances engines, airframes, etc. Instead, the Technical Standard Order, or TSO is issued as a generic standard to which each headset must comply.
TSO C50c is derived from Radio Technical Commission for Aeronautics (RTCA) Document No. DO-170, which prescribes standards for everything from audio selector panels to headsets. Additionally, all parts except knobs and switches must be in compliance with the flammability requirements of 25.853, 25.1359, and Appendix F of Part 25. (Folks sometimes get confused as those parts deal with aircraft interiors, but the TSO references them as a requirement in compliance with the TSO...which is a common practice in regulation and standards). Your headset must meet flammability and flame propogation standards...something most people have probably never even considered.
The same RTCA standard is also used to establish TSO C57a, which specifically provides the standards for aircraft headsets. All headset manufacturers seeking TSO recognition of their product must meet this standard. TSO C58a sets the standards for the microphone.
These TSO's are in turn governed by Part 21, subpart O, which is the regulatory teeth for establishing the need and compliance for TSO's.
The bottom line for you, all having been done and said, is that if the aircraft has ample other headsets or speakers on board to meet the requirements of the part under which you're operating, you can supply your own headset. It's a personal item, not an aircraft item...just as you can supply your own pen, coffee mug, and toothbrush.
Where you do need to particularly careful is how that headset will interact with the aircraft cockpit audio system and intercom, as well as communications in general. Mixing headsets and microphones of varying impedences can wreak havoc in a communications system. I've seen very sophisticated, very expensive ICS systems made inoperative merely by using different headsets. One location where I worked, we solved the problem by hardwiring the headsets in place; crews were bringing their own and it was causing a lot of problems. We supplied everything David Clark (could have been any other type, too, but these we knew to work well with that system). It could be the headset causing problems for you, or the mic causing problems for others...but be sure your headset is compatible with the rest of the system.