I was an avid tea/coffee drinker prior to this pregnancy --- morning sickness has turned me off to so much though!
I don't think one cup of smooth move would hurt but it's probably not best to rely on it --- especially if it has licorice root/senna --- those should only be used over a short period and in moderation and are still not actually recommended.
I've researched the properties of quite a few different herbs (books regarding tinctures and tea) and have consulted two naturopaths/herbalists for further information in regards to what's "safe/beneficial" during pregnancy --- I concluded that dandelion tea, nettle tea, rooibos, peppermint, ginger and raspberry leaf (in the third trimester) are safe enough for me to drink.
Dandelion tea is a mild laxative and also contains many vitamins and minerals (whilst supporting the liver) --- something you may want to try. In conjunction to that, I'd recommend a diet of fruits and veggies (If you can keep it down! You could make a smoothie out of blueberries [lots of fiber], carrots, vanilla yogurt/frozen greek yogurt, milk substitute [almond etc], ground flaxseeds, orange juice/orange sherbet [naughty but tastes good lol], for example). Moderate exercise (again, if possible! I know how hard it can be) will stimulate bowel movement --- try walking after every meal if you can. There are yoga exercises that can also aid bowel movements (might be worth looking into any classes during pregnancy around your vicinity). Keeping hydrated is a necessity too... but can also be an obstacle if you're ill.
Warm lemon water (and soda omg lol) is what personally got me through most of my morning sickness. I've only recently begun introducing more drinks, including teas.
And I drink this raspberry leaf tea that's supposed to be good for menstruating or pregnant women 
