Your question is very similar to Motion of a bouncing sphere with a spring attached inside.
As you have realised, the time between bounces is affected by the motion of the platform, and vice versa. If the platform is moving upwards (downwards) when the ball collides with it, the relative speed of impact is greater (lesser), so the impulse is greater (lesser), and the ball rebounds higher (lower) than before. The time between bounces therefore varies, as well as the impulse delivered during each bounce.
In general the resulting motion is chaotic. However, there will be some special cases in which the particular combinations of the values of the parameters $m_1, m_2, k, c$, and the height $h$ from which the ball is released, result in periodic motion. You could look for these by specifying that $F(t)$ is a periodic 'kick' of constant value, and finding the condition on the parameters for this to be a steady-state solution. (See notes cited below.)
However, in general it is not possible to define the force $F(t)$ because it depends on the unknown behaviour of the platform, which likewise depends on the behaviour of the ball. You cannot know the form of $F(t)$ until you have solved the equation of motion which contains $F(t)$, and you cannot solve that equation without knowing beforehand what form the function $F(t)$ takes.
There are 2 ways round this difficulty. The 1st is to simulate the motion numerically, in small time increments. The 2nd is to approximate, for example by making the assumption that the motion of the ball is not affected by the motion of the platform - then $F(t)$ is a periodic impulse of constant amplitude (see for example section 7.2 of these notes) - or that the motion of the platform is not affected by the motion of the ball.