There's a simple mathematical argument as to why this might not be as useful as you'd think, even given a really cheap means of acceleration.
Let's say we have a relativistic "particle". It could be a fundamental particle or a space ship or a planet, doesn't matter. The time which passes in the lab frame is given by
$$
T = \gamma T_{0}
$$
where $T_{0}$ is the time that's passed in the particle frame, and $\gamma$ is
$$
\gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}
$$
So if you have a cheap accelerator you can just make $v$ huge, and make $\gamma$ like 10,000. Now your radioactive thing lives 10,000 times longer, right?
Well, maybe that's not worth your time. The energy of your particle is given by
$$
E = \gamma m c^2 = \underbrace{(\gamma - 1) m c^2}_{\rm kinetic} + \underbrace{m c^2}_{\rm rest}
$$
where I broke the terms apart to show the rest energy and the kenetic energy. So for $\gamma \gg 1$, the kinetic energy is much larger than the rest energy.
The important part is that you can create your particle from thin air if you can produce the rest energy*. So for the energy that you used to extend the lifetime by a factor of 10,000, you could have created 9,999 of the same particle (by smashing some particles into a target with an energy equal to $m c^2$, for example).
This isn't to say that this would never be worth it: maybe the process to create your particle is really inefficient, and accelerating it is really cheap. But in general you're battling against the same factor of $\gamma$ that you're using to dilate time.
*Creating particles from thin air is subject to some conservation laws: you might have to create some byproducts in the process.