Considering that, I don’t see why the internal energy increases. Is it
because the increase in kinetic energy exceeds the decrease in
You haven't specified the details of the process, but if in fact the internal energy increases as a result of the compression while the internal potential energy decreases, then yes the increase in kinetic energy must exceed the decrease in internal potential energy to satisfy conservation of energy (change in internal energy = change in internal kinetic energy + change in internal potential energy).
In view of your recent edits, namely that there is no heat exchange with the surroundings and the gas is non-ideal, then there will be an increase in internal energy. For that to be the case, the increase in internal kinetic energy due to the compression must exceed the decrease in internal potential energy in order for there to be an increase in internal energy. So my answer still applies.
Hope this helps.