So you have a body travelling at such a speed that it's $\gamma$ factor is non-negligible. How is the mass going to become manifest anyway? One way would be through its gravity: there could in principle exist a balance so big that it's set up in a uniform gravitational field, and the moving body is on one pan, and there is an identical stationary body on the other, and the pans are so big that despite the near-luminal speed of the moving body we have time to see how much mass needs to be added alongside the stationary one to balance the balance; or maybe we could have two bodies initially coalesced, and two (two to obviate wobble) test bodies orbiting about them, and then they are propelled apart symmetrically at near-luminal speed, and the mass inferred from the change in the motion of the test bodies: or maybe we could infer the mass from momentum by colliding the moving body with something, or by some how measuring the reaction in whatever it was that propelled them: or through energy in a manner similar to that whereby we might reach it through momentum (particles created in a collision in an accelerator, that sort of thing): or ...
Yes I realise the foregoing is a highly laboured disquisition; and also that much of it could not actually be performed by human: but it is intentionally laboured, as I am trying to demonstrate that beyond classical dynamics, mass is essentially inseperable from the way in which it is manifested. You can manifest it through momentum, in which case the immediate datum is the behaviour of momentum in the relativistic regime, or similarly through energy; or you can manifest it through gravity, in which case the immediate datum is the behaviour of objects under gravity when they are moving at near-luminal speeds. In no case is mass itself the immediate datum. (I am being a bit pedantic here, BTW, and using the word 'immediate' perfectly literally to mean without mediation rather than right now). Mass as immediate datum is really a classical concept, proceeding from being able to actually have the stuff there, examinable & palpable, and which has installed itself as a habit of thought that we could do with breaking when we move onto the relativity paradigm, as by the very stipulating of the moving frame of reference we are setting mass per se out of reach.
But then I'm considering motion in a straight line of course. I cannot find any straight answer to the question of whether a particle moving in a circle has a Lorentz factor on its mass. I think in that case the entire system of kinetic energy of particle + its potential energy in whatever field is constraining it to circular motion must be considered as a whole. But we know that in nuclear reactions, the difference in mass of the substance from ante -reaction to post -reaction is equal to (energy emitted)÷c²: that is very thoroughly established experimentally; and we are talking about mass considered in an altogether unextraordinary sense - weighable in a balance. Both the potential energy of the binding of the particles together, and the kinetic energy of the motion of the particles around each other are making contributions to the total mass. And some of that potential & kinetic energy can be extracted as energy in the usual sense; and the total mass is mass in the usual sense. So it would seem from this that maybe relativistic mass is not altogether merely an expedient abstraction.
So a body moving in a straight line at relativistically significant speed is not practically weighable, although fabulous and yet physically realistic - though not humanly implementable - means can just be devised for (mediately) weighing a body moving at relativistically significant constant velocity in a straight line; whereas subatomic particles moving at high speed within a piece of matter constitute a system in which the contributions to the total mass of the kinetic energy of their motion and the potential energy of their being bound together cannot simply be extricated from each other in the relativistic régime ... although the system as whole may well be simply weighable.
Looking for particularisations of this constrained non-linear relativistic motion, the only two I can find are essentially also quantum mechanical: the first one - the atomic one, is that of motion of electrons in an atom. This is dealt with by the rather difficult & complicated Dirac equation, although approximate results can be treated by setting relativistic effects as a perturbation. The perturbation approach is reasonable inthat for a hydrogen atom the motion is only barely relativistic - the $\beta$ factor having a value equal to the fine structure constant $\alpha$. Whichever method you take, the result is an expansion in terms of $\alpha Z$, $Z$ being atomic number. Or $(\alpha Z)^2$, really, as the terms tend to be even. According to this, relativistic effects become the more significant the higher up the periodic table, and account for certain anomalies that set in, such as gold being yellow, and cæsium being more electropositive than francium; and also for the possibility that an atom cannot even exist with $Z>137$, as the mentioned series would diverge for such an atom. (Some, however, say that this effect might be circumvented, and that this argument does not in fact preclude the existence of such atoms.)
But the point here is, that in treating of an atom thus, there ceases to be any equivalent of the gorgeous virial theorem, whereby the total energy is equal to the negative of the kinetic energy and the potential energy is twice the total energy - this kind of segregation sloughs away, and it's all framed in terms of total energy alone. So the relativistic mass of the electron, & it's corresponding energy, becomes thoroughly conflated with the total energy of the system. The other system which is that of the quarks constituting baryons: the motion here is certainly ultrarelativistic but there is certainly no more talk about the 'shape' of the quark's 'path'! All that is longsince gone! You might well read that the restmass of the quarks (and you can't even have free quarks anyway! whence there is controversy over even defining the restmass of quarks atall! ) is a small(ish) fraction of the mass of the baryon, and that it's mass is very much preponderantly due to the energy of the quarks' binding & motion. So here we have relativistic mass being very palpable - but alongwith a thorough foiling of any hope of attributing it to a nice Lorentz factor with a nice $\beta$ & $\gamma$, & allthat.
Is the motion of particles in a cyclotron or synchrotron a sufficiently tight circle for the departure from the approximation of linear motion to be significant in this connection!?
I do see how the notion of relativistic mass is a very heavily fraught one - but on the other hand it does not seem altogether to be one that can be utterly dismissed.