the concept
I think your problem may be coming from a misunderstanding about when you can apply conservation of energy and momentum.
In fixed inertial frame energy and momentum are conserved separately, so $E_i = E_f$ and $\vec{p}_i = \vec{p}_f$. If you select the rest frame of the initial particle as your working frame for the whole calculation, you can use this fact.
When using different reference frames for the before and after pictures, energy and momentum are not conserved separately. Only the magnitude of the total energy-momentum four-vector is: $\tilde{p}_i\cdot\tilde{p}_i = \tilde{p}_f\cdot\tilde{p}_f$, where $\tilde{p}\rightarrow(E/c, \vec{p})$ is a four-vector.
The four-vector statement is where your energy equation comes from:
$$\tilde{p}\cdot\tilde{p} = -\frac{E^2}{c^2} + p^2.$$
Working in the the rest frame of the system in question, $\vec{p}=0$ and $E = m c^2$:
$$\tilde{p}\cdot\tilde{p} = -m^2\, c^2.$$
Finally since $\tilde{p}\cdot\tilde{p}$ is the same in all reference frames:
$$E^2 = p^2 c^2 + m^2 c^4.$$
Applying it to your problem
To start your problem you need to pick a reference frame for each state, before/after decay. Lets pick the rest frame of the initial particle, so
$E_i = m_1 c^2$ and $\vec{p}_i = 0$.
If we use the same reference frame for the final state, we can set up conservation separately. After the decay particle 2 is moving so $E_2 = \gamma m_2 c^2$, where $\gamma$ is the Lorentz factor. The massless particle 3 has energy. We can get it by applying the energy-momentum equation to it, alone:
$${E_3}^2 = {p_3}^2 c^2 + 0$$
So
$$E_i = E_f \implies m_1 c^2 = \gamma m_2 c^2 + |p_3| c.$$
We still need particle 3's momentum, but we can get that from momentum conservation:
$$\vec{p}_i = \vec{p}_f = 0 \implies \vec{p}_2 = -\vec{p}_3.$$
All of the motion happens in one dimension, and $p_2 = \gamma m_2 v$
From here we can put it all together and solve for the speed of particle 2 (or $\gamma$ if you prefer).
