Is a fitting worthwhile for me?
My advice is that you consult a sports medicine specialist, for example a physiotherapist, especially one who specializes in knees.

Once you find such a person, find a second. Sometimes you just don't get along with them. For me, I have to feel I can trust the person, and get honest discussion of the options and the pros and cons.

While bike fit may be an issue, it's only one approach. Other things to be considered are


gearing. Perhaps you just need to pedal at a higher cadence.
power and speed. Perhaps you're just trying to pedal too hard for that knee.
technique. Perhaps pulling up more would reduce the load on that knee.
crank size and cleat positions. These can both have an effect on knees.
warmup and cooldown. Getting symptoms at the start sounds like lack of warmup.


Find such a sports medicine specialist, and work with them as they evaluate your options and manage your progress.

Following your edit here are some additional points


wobbling pelvis and pain at the back of the knee are both symptoms that your seat is too high. 
just as an idea, you could try a topical warming agent (ointment) to help with the warmup. 


In an earlier version of this post I suggested Voltaren, recommended for me by my physio. I think that was unwise of me - you should obtain professional medical advice. Voltaren is an anti-inflammatory drug, available under prescription in the US. Some researchers warn against it's use.