You can't increase the rest mass without putting more stuff in, but you can increase the effective mass, yes.
For example, the mass of a capacitor becomes ever so slightly greater when charged. How much? So little, the natural diffusion of atoms (not just of air, but even seemingly inanimate ones) around the capacitor utterly dominates the change in mass -- that is, if you could measure the exact mass of molecules bound to the object-as-such, you would find the signal has orders of magnitude more noise.
It's worth noting that special relativity is an approximation -- there's good reason why it's always "inertial frame of reference". An accelerated frame of reference is not relative, and all sorts of quirky things can happen. Apparently, higher order terms can lead to inexplicably unbalanced acceleration or energy (force?). But, as the OP suggests, these terms need relativistic velocities and accelerations to become significant. (Higher order terms being what they are, they drop out very quickly indeed, at low, non-relativistic accelerations.)
So it's not a bad idea. Perhaps a good idea, even. It may be analogous to a heat engine, how compressing and expanding (and heating and cooling) a fluid allows power to flow and work to be done. Well, suppose we had some means to compress and expand space itself. Perhaps we could displace the space while it's compressed, and as a result end up moved further along without apparently having touched anything (Mach force). Subject, of course, to analogous limitations -- thermodynamics won't let you go that easily.
Some have done experiments attempting to show just the subtlest hint of these higher-order terms; a relativistic elliptical whirly-go-round isn't exactly feasible, or constructible of even the strongest normal matter (electron-orbital bonds are a few eV per atom; that would need 100,000s, or 1,000,000,000s, of eV to be possible; hence the tease about neutronium). These usually take the form of a coupled vibrating-capacitor device, so that the capacitor is accelerating the most when it also has the most charge. The acceleration is provided by a piezo stack, which also stores energy; an inductor balances the electric circuit so that only losses need to be provided to the system.
So far, at best micronewtons have been detected, and that's using "detect" generously as this is the detection limit of a microbalance. For forces this small, any and every possible interference must be accounted for -- forces as subtle as the emission of adsorbed gasses from one side of the apparatus more than another, greatly exceed the signal being looked for!
Relating to another somewhat fringe idea, it's expected that this mechanism would serve as the "exotic matter" of the Alcubierre warp drive. Now we just need to find piezo disks big and strong enough to contain a small planet's worth of mass-energy...
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Tim