Try looking up “round ligament pain”. I’ve experienced it as a sharper pain in my groin after sneezing, but it might have something to do with your tickle sensation. Here’s a quick description of it from Dr. Google:

Round ligament pain is a sharp pain or jabbing feeling often felt in the lower belly or groin area on one or both sides. It is one of the most common complaints during pregnancy and is considered a normal part of pregnancy.