PID controller quickly drops command signal and quickly rises it when initially given control?
The sharp drop is because the error is high. The proportional term is going to go way down, and since you have no integral built up, of course the output will drop. 

I'm not exactly keen on the "lookup table" approach, I'd rather see the loop better tuned, but if you are okay with the lookup table, just force that value to the integral term, suppressing P and D until the time expires, then enabling the full PID with the integral preset.

With regard to the comment about increasing the proportional gain, I'd advise against that as increasing the proportional gain would probably drive that system into oscillation; and increasing it would also cause a much sharper drop when switched from "Startup" to "Run" mode.

Ultimately, you probably do want to pre-load the integral term with your pre-calculated output value and just let it run.  Tuning PID's by just playing with the gains (or gain/reset/rate, depending where your come from) usually doesn't end up with a stable system.

There are many methods of tuning PID loops, but if you must "Wing" it, typically reduce the I and D terms to zero, run P up until the system oscillates, and back P off by about half.  That should give you a steady state error.  Next, increase I slowly, until the error is eliminated, and the system doesn't hunt.  D should normally only be used slightly in most loops, but might be necessary in very dynamic controls.