Now, we also know that an accelerating charged particle radiates electromagnetic energy. In this case, we would have synchrotron radiation.
We know it radiates EM radiation, but whether there is energy flow associated with this radiation is a more subtle question.
We know with certainty radiation of groups of very many charged particles that move and accelerate together, as a quasi-compact system, carries away EM energy, and this $group$ is continually slowed down and to continue moving the same way (with the same amplitude), has to be re-accelerated by external force (due to driving circuit of an antenna, or acceleration cavity in synchrotron). In case of synchrotron, the group of particles moving together is called bunch.
In case the charged body/particle is composite, made of several charged parts, we can regard it similarly, and the radiation carries EM energy a la Poynting.
For radiation of single point particle in magnetic field, we don't know of a consistent theory where such radiation carries energy, at least not in the usual sense. The Poynting theorem cannot be interpreted in terms of energy in case of point particles ($\mathbf j\cdot\mathbf E$ is undefined, total Poynting energy is infinite).
There is a class of mathematically consistent EM theories of point charged particles (the Frenkel/Feynman-Wheeler type of theories), which assigns zero energy/momentum flux to field of a single point particle, and non-zero electromagnetic energy/momentum density, positive or negative, is associated only with pairs of fields (of two point particles).
Electron is a point particle in the Standard model, but we do not know whether electron is a point particle or not in reality; the electron could have some minuscule size. Thus we also do not know whether radiation of a single accelerated electron in magnetic field carries away EM energy a la Poynting. All experiments observing such radiation ("one electron cyclotron") actually seem to observe radiation due to very many particles (currents in the walls of a waveguide, particles making up the experimental setup).
There is the adapted Feynman argument from gravity waves (the EM wave moves the charged body sliding on a rail so the rail heats up and thus the EM wave has to carry the energy), but this actually implicitly assumes that that energy had to come in with the wave and could not have been just extracted from the local EM energy in the system. The latter is actually the case for EM wave due to single particle in the Frenkel/Feynman-Wheeler type of theories, so the adapted Feynman argument is actually invalid there.
Your result $|\mathbf v|=const$ is valid only if the simplest form of the Lorentz force law for point particle is used:
$$
\mathbf F = q\mathbf E_{ext} + q\mathbf v\times\mathbf B_{ext}.
$$
If the particle is made of (two or more) charged elements, this is not the entire force acting on the system, because now the elements act on each other with internal forces, and in special relativistic theory, net result of these internal forces need not be zero force. For a uniform charged ball/sphere, Lorentz and Abraham derived that there is additional force, sometimes called self-force, so the total force is approximately
$$
\mathbf F \approx q\mathbf E_{ext} + q\mathbf v\times\mathbf B_{ext} - m_{EM}\mathbf a + \frac{2}{3}\frac{Kq^2}{c^3}\frac{d\mathbf a}{dt}.
$$
where $m_{EM}$ is so-called electromagnetic mass of the ball/sphere (of the order of electrostatic potential energy of the system/$c^2$), $K=\frac{1}{4\pi\epsilon_0}$ and $\mathbf a$ is acceleration of the whole charged ball/sphere.
The EM mass term is written with minus so positive EM mass increases the effective mass of the system (this is the case e.g. if it is made of same sign charges, or at least if total electrostatic energy is positive), and the last term is the so-called Lorentz-Abraham-Dirac force, or radiation-reaction force.
This changes the analysis and makes it possible that the accelerated charged body actually loses energy in magnetic field.
... $U$ would no longer be the energy of the particle, but instead would be the energy of a new system: the particle + the radiated field (let's say "a collection of photons"). Now, the energy of this system would be conserved, as Lorentz law says.
Total energy of isolated region would be conserved only if there is zero energy flux on the boundary of the region, so the boundary has to be put far enough and there must not be acceleration in the past. This is pretty obscure and hard to check, so it is much safer to talk only about local conservation of energy: this means that we do not say that total energy of particle + EM energy is conserved, but only say that this energy can change only via EM energy flux through the boundary. But this law of local conservation of energy is a general result, going beyond the specificity of the situation where magnetic Lorentz force does not work and is the only force present.
