If a particle falls into a black hole from a great distance, when it reaches the singularity (assuming it doesn't collide with anything) should have the energy required to escape.
Wrong. In general a particle falling on a gravitational well, the earth for example, from far away can have a very large velocity, but whether it will collide, be captured as a moon, or escape by scattering off the gravitational field depends on the particular angles and velocities etc, and equations have to be solved.
However, escape requires a velocity in excess of the speed of light, how can that be?
This statement does not apply to an incoming projectile on the black hole, see the previous paragraph. It applies to hypothetical particles already within the black hole's horizon, where General Relativity has to be used for the mathematics,
Intuitively to me, it seems that the particle should gain mass as it approaches the singularity,
Relativistic mass is not a helpful concept , it is better to think of increasing in-falling momentum, supposing the newtonian gravitational well solution gives an in-falling trajectory
and then lose that mass on it's way out and escape.
Only trajectories that are not on a collision course , hyperbolic or parabolic, behave like that , do not fall in and scatter off back to infinity, losing the energy gained while falling into the gravitational well.
But that only raises more questions, like why can't any particle pass though a small (volume) blackhole's event horizon?
Black holes are a mathematical proposal from the theory of General Relativity and have been detected in astrophysical observations. It needs the mathematics of GR to understand what is happening, which cannot be handwaved away. Once caught below the horizon the particle cannot escape is what the solutions say.
In everyday Newtonian gravitation, a chemical reaction can be used to escape from a potential well by providing enough energy, that is how we have satellites . In GR the mathematical solutions are such that no amount of energy will allow escape from inside the horizon. This hyperphysics link gives a thread to follow the mathematics of the case for black holes..