hi there and thanks so much for posting this interesting question in my forum!
In my experience working with pregnant woman I never experienced that lower back pain could associated with contractions but that doesn't mean that what you are describing is accurate.
According to a woman after the 21st of pregnancy and experiencing similar effect this was her answer: It will continue. I'm a mother of 2 so I know. Get a heating pad and use it on your back. Always lie on your left side when sleeping,the blood flows better that way and alleviates pain to an extent. copy and paste the following link http://*****************/question/index?qid=20080108084621AApGALq
Also check out this great article from James W. Brann,MD copy and paste the following link> http://www.womenshealthcaretopics.com/pregnancy_week_22.htm
From your case description I would like to suggest the following. Normally when you are pregnant you should follow a similar activity than before the pregnancy if you decide to add some sort of physical activity you need to be extremely careful especially in the changes of hormonal activity. Strength training with weights adds extensively changes in hormonal activity. So if you were not lifting weights before and you are doing weights you need to be extremely mild I would stay away if you were not doing it before.
Second lower back pain in pregnancy is associated with lack of core strength also especially if you had a weak core before the extra weight produces an impact in the lower back where our gravity center is located. That happens in animals and humans right now we know that we need to transfer and increase in strength in the core where the lower back it is just a part of the core. What are the best core exercises? those ones requiring balance like one raising your heels or lifting one leg as long as you are safe.
My suggestion is that you keep working with your doctor really closely and with your trainer and maybe adjust the activity to where you don't feel that uncomfortable pain. Add breathing exercises, stretching and core exercises such a popular core bridge exercise.
I hope this answer was helpful and I am sure everything will be fine. Let me know if you have anoy other questions