The important things to take from the question:
1. elementary particles could have an intrinsic/quantized rest mass
2. Any rest mass that is not a whole number multiple of this constant has to be attained from interacting with the Higgs Field
Answer
You have just described exactly how things are (to current physics knowledge) in our own universe, with the quantised mass-constant $= 0$, where any non-zero mass (of the fundamental/elementary particles) is obtained via interaction with the Higgs field/boson. While all fermions attain a mass via the Higgs, not all gauge bosons do (the photon).
If, however, you want a non-zero mass-constant, then this requires breaking charge conservation (electric, weak and strong/colour). Stability is more to do with the total mass, with the decay constant of a particle related to the imaginary part of the mass of that particle.
Explanation
So, where does mass come from (apart from the Higgs)? (here, intrinsic mass will be used to refer to mass that isn't obtained from the Higg's mechanism)
That would be the Standard Model of particle physics (Brace yourselves...): all the known elementary particles and forces (excluding gravity) are described by terms in the following: $$\mathcal{L} = \mathcal{L}{Higgs} + \mathcal{L}{gauge} + \mathcal{L}{lepton} + \mathcal{L}{quark} + \mathcal{L}_{Yukawa}$$
Each term in the above contains mathematical terms such as $\bar f\cdot \gamma^a g_2 W_a f$ (in this case, interaction of a lepton or quark with part of the electroweak field. Other mathematical terms are not written for brevity), which can be drawn as part of <PERSON> diagrams, which in turn gives all the maths/statistics required to do particle physics. Including calculating the mass of the elementary particles - a term giving a particle $p$ a mass is written as a term such as $m\bar pp$.
From condition 2, we can ignore $\mathcal{L}_{Higgs}$.
What about the other terms? $\mathcal{L}{gauge}$ contains the electroweak and strong fields, which have no intrinsic mass. Actually, neither does $\mathcal{L}{lepton}$ or $\mathcal{L}{quark}$. 800
So, we're left with $\mathcal{L}{Yukawa}$
$\mathcal{L}_{Yukawa}$ is, of itself, still quite long, containing terms involving the allowed possible interactions of fermions with the Higgs boson, so can also be ignored by point 2.
Or, the only way that elementary particles can have mass is from the Higgs boson. This includes, to my knowledge, neutrinos.
The reason that this is how things are is that any possible mass terms not arising from some form of interaction with the Higgs in the standard model wouldn't be invariant under what's known as gauge symmetry. Due to what's known as <PERSON>'s theorem, this means that making the Standard Model non-invariant under gauge symmetry would break conservation of charge - electric, weak and strong (colour).
Putting it another way, the only way that your top paragraph isn't true (in a universe where the standard model is correct, anyway) is by not having charge conserved. If charge is conserved (and the standard model is correct), then your first paragraph is true, with the constant being $0$.
Conveniently, the Yukawa coupling gives all fermions an 'attained mass'. Also convenient is that the photon, a gauge boson remains massless. However, no bosons have 'only intrinsic/quantized mass with no attained mass' [I've removed the word 'rest' because I'm pedantic about not using the term 'rest mass'] where the intrinsic mass is non-zero (the photon is a gauge boson with zero mass). If you want this to be the case, then feel free to violate charge conservation and the set the mass-constant to be whatever you like!
How stable is an elementary particle?
For a non-rigorous thought about how decay of a particle works, let's imagine that we've got a stationary particle* of mass $m$. This gives the energy as $E=mc^2$ and its time evolution is described by the <PERSON> equation $i\hbar\partial_t \Psi = \hat{H}\Psi$. 800