Fundamentally, the source of the temperature increase is not the piston colliding with the gas particles: it is a result of the same (molar) amount of gas, with the same amount of total energy, but now in a smaller volume.
The integral of piston motion force times distance $F\cdot d$ (or more commonly the integral of pressure times change in volume $P\cdot dv$), work done, is merely a useful way to account for the work required to reduce the container volume in the absence of heat transfer through the container walls. With or without colliding with the piston, the gas molecules will heat up. Why, they have a higher energy per unit volume when the volume is decreased.
Really so? Imagine you have a container with molar amount $n_1$ of a gas at pressure $P1$ adjacent to (but isolated from) a larger container with amount $n_2$ of the same gas at a higher pressure $P_2$ and isolated from surroundings (fig below). The gas in the larger container is at a higher temperature $T_2$ due to its higher pressure.
Now, open a valve between the larger container and the smaller one, so that the pressures equalise. The mixed pressure $P_3$ will be between the original $P_1$ and $P_2$, and the mixed temperature $T_3$ will be between the original $T_1$ and $T_2$. The original amount $n_1$ of gas now occupies a smaller volume since it now shares space with some of the gas molecules that moved in when the valve was open (figure below - right hand boxes).
If you used a piston to compress the gas in the smaller container, of amount $n_1$ at $T_1$ and $P_1$ (boxes on left in figure below), for the same final temperature $T_3$, the final volume is the same volume that the gas would occupy when compressed by the incoming gas (boxes on the right in the figure below). $P_3$ is also the same.
It matters not how you got the gas molecules to occupy a smaller space: if total energy is not lost from the gas, the molecules will heat up due to the increase in mutual collision frequency. In other words, the gas has more energy per unit volume.
Isothermal compression without energy transfer is not possible given the conditions you state.