OK I am assuming the Dolce fits you OK and you are not thinking of upgrading because it doesn't.

IMO the answer would be No.
I have ridden aluminium and carbon bikes, however my AL bike was a hybrid (and was heavy) and my carbon was a Ruby (actually half the weight). The Dolce isn't hugely different to the Ruby.

There isn't a huge difference in fit, riding style or weight, so you are not going to get a huge speed advantage. Yes the ride will be a bit more forgiving but it's not an overwhelming factor.

Riding is really how you feel on a bike and how much the rider does or improves. If you reasons for upgrading are to go faster, well to be honest you really need to ride more to do that. A new carbon bike probably isn't going to help that much.

I think that to ride more, it would be more of a committment issue for you to work out, eg getting up earlier and going for a ride, finding child care, joining a cycling group etc. I don't think having a more expensive carbon bike will make you ride more- it sounds like it could be more stressful as you will make yourself feel guilty that you are NOT riding it as often as you feel you should.

However I'm hugely impressed that you DO ride with 3 children. The number of people who have left my cycling club after they have a baby is huge and I don't ever see them again which is sad...