I think the confusion is treating everything constant between power, force, and velocity, but this isn't possible.

If power is constant, then $P=Fv\to F=P/v$, meaning $F$ is a monotonically decreasing function of $v$. So when $v$ is "small", $F$ is "large". This causes an acceleration, which increases $v$ and decreases $F$. Note that this has to happen for constant power. There are no other forces present. You have said the power is constant for this process (whatever it is), and so the force has to be a function of $v$ as $F=P/v$.