Find optimal gains transfer function
If you specify a rise time, then you must also specify the working frequency. It could be deduced as \$\omega_0=(200K)^{1/3}\$. Saying K>0 means no clear information, it could be as low as 1f (or lower), or as high as 1T (or greater). Which means that the 0.5s time is tied to \$\omega_0\$. Once you have it, you can use either some optimization program, or some simulator of choice to step the value for Kv, then manually determine a "best" value.

I would have just made this a comment, but it turned out a bit more. Here's, for example, what a simulation for K=1 means: Kv>0.025 so step the value from 0.05 to, say 0.25:



with a step response like this:



Since there's no automatic tuning, zooming in over the responses reveals that
the best response for an imposed settling time is between the pink (5th, Kv=150m) and grey (6th, Kv=175m) traces:



Stepping with a 5m step reveals the best values are between Kv=160m and Kv=170m:



And, finally, with a 1m step:



The green trace seems to be the one, which is Kv=163m. The tuning can go on from here, if needed.