The force responsible for fission or fusion is the "strong nuclear force". I don't have the numbers readily at hand, but the strong nuclear force exceeds what we normally encounter in the physical world by many, many orders of magnitude. This causes the very small mass converted into enormous energy in the case of a nuclear bomb to be readily measurable. But any of the physical forces operate similarly.
Let's try another thought experiment.
Imagine a proton at rest relative to you, the observer. If you apply a slight force to the proton, it will accelerate. Newton's Second Law dictates that the aceleration will follow F=ma. That is, for a given mass, a slight force causes a slight aceleration.
If not for Einstein's discovery, Newton's Second Law would dictate that the aceleration could continue at the same rate as long as the force remained constant. If we push on the proton long enough, it should exceed the speed of light eventually.
Einstein's discovery of E=mc2 modifies the situation so that the speed of light can never be exceeded. Instead, what is observed is an increase in mass as the velocity increases. F=ma still applies, but the observed mass becomes greater than the "rest mass".
The observed increase in mass is exactly what is required to account for the energy stored in the moving proton. Energy added to the moving proton can be calculated by multiplying the force applied times the distance through which the force acts. This works out to mv2/2. This is called the kinetic energy of the particle, where m is the observed mass and v is the velocity.
At low velocities, energy added to the proton causes an increase in velocity and the observed mass is near the rest mass. As velocities are reached which are near the speed of light, additional energy added appears as an increase in observed mass and the velocity changes only slightly. The observed mass grows sufficiently large as the speed of light is approached that the force cannot cause the velocity of that large mass to exceed the speed of light.
That's where you're wrong. They exceed the speed of light in almost every Star Trek episode ever made.