The answer is yes any way you slice it.
However, if you try to quantify it or measure it, then you'll get different answers depending how you slice it. So let's see how the slicing matters.
Firstly, you asked about mass, but then tagged it with mass-energy. And mass and energy are different. In particular mass is associated with rest energy but there are lots of other kinds of energy. And historically there was a pish yo try to use the words relativistic mass as a synonym for energy and that tuned out to be a bad bad idea. Just say energy if you mean energy and just say mass if you mean mass.
But this leads yo the second issue. The mass of a system isn't the sum of the masses of the parts. Your wire might be made of electrons and protons and neutrons and gluons arranged in a metal with a shape, each electron and proton and neutron can have its own individual mass but the mass of an atom in the metal is not the sum of the masses of the electrons, protons, and neutrons that make it up. And the mass of the metal isn't the sum of the masses of the atoms and such that make it up. That's life.
And mass is really just how energy and momentum are balanced, e.g. through $E=+\sqrt{p^2c^2+m^2c^4}$ where $E$ is the energy, $p$ is the magnitude of the momentum, $m$ is the mass and $c$ is a fixed universal constant.
Since the magnitude of the momentum of a system isn't the sum of the magnitude of the parts, the mass of the system isn't the sum of the masses of the parts. And effectively the mass is like the length of a vector, when a bunch of vectors point in mostly the same direction the length of the sum is really close to the sum of the lengths. And that accident is why mass seems additive when things have a relative motion to each other that is slow compared to $c$.
And what's further complicated is that some things you think are caused by mass (such as gravity) are actually caused by energy but you are used to energy being dominated by rest energy and rest energy is proportional to mass.
So its tricky to point at something and say what's the mass of that. Because are you trying to divide up a fair share of a total mass of a system or are you trying to find out how energy and momentum are balanced amongst just that part. Both are valid things you might what to know. Since mass says how energy and and momentum are balanced you want to know that. But when a system allows the momentum of different parts to cancel that can make the mass of the system larger than the sum of the masses of the parts.
But its worse. You can have interaction energy, so the energy of a thing by itself and another thing by itself might not add up to the energy of the things together. Plus what are the parts anyways?
In a wire you have ions, mobile and stationary. And atoms, And you have the fields generated by the wire. Each of those has their own momentum and energy and hence mass. But the system also has a total energy and total momentum and hence a mass for the system.
And in this case the system includes the fields. And the fields are everywhere, they fill all of space. And there is one field, whose values is partly determined by everything in the universe. So what parts get to own it?
And this is where lots of the energy goes when you turn the device on. Some energy goes to increased kinetic energy of the mobile ions such as electrons in the wire. And some initially goes into the fields right by the charges as the charges underwent a mediated interaction with each other through the fields. But that energy started to move around as time went on. And some of the energy originally given by some charges in some parts of the wire got passed back to other charges as the current ramped up.
So we saw energy go into the fields but we can't just add up all the energy that went in because some energy came back put of the fields and if we try to add up all the energy in the field we'd start adding up energy in distant galaxies far away which is a bit silly. But there isn't a natural place to stop.
In introductory classes we can model a capacitor as two plates with fields only between the plates. And we could model am inductor as a coil with magnetic fields only inside the coil. But both are wrong. The fields are sometimes a bit weaker than that and some of it spills out but having it spill out means it becomes hard to says whose energy that should be.
So the problem is that it's hard to know which energy exactly you want to measure. If you can't say what you want to measure, then it is hard to make a torsion balance or something to measure it precisely.
So the basic idea of having a powerful coil and having it be more massive comes down to having it have more energy without a corresponding increase in momentum.
There is definitely some energy going in. But it is not much energy compared to how much energy was already there. So the mass isn't changing much. So a small effect and a vague formulation means there isn't much hope for a rigorous confirmation without a specifically designed test. Which is hard with a vague formulation.
If you have reasons to doubt specific things, then you might see that those doubts have already been studied and addressed. But you'd have to be more specific.
