Hmmm, I was going to say “Go for it!”, but some of what you say makes me think twice.
From the perspective of health of the baby—You won’t cause any harm, after all humans colonised the world while some of them were pregnant. If our ancestors had stopped walking every time one of them got preggo, we’d never have got beyond the Rift Valley. As walks go, the Camino is pretty gentle—you aren’t mountaineering or putting yourself through temperature extremes; and you will never be far from other people or facilities. If you’re staying in any kind of serviced accommodation (ie not camping), you don’t need to carry much. And exercise is good for the baby (via the mum’s circulatory system) as well as the mum.
The thing that makes me hesitate is what you say about your symptoms, both in terms of impact on you but more particularly on other people. If you are still vomming/pooing when it comes to the walk, it frankly doesn’t sound like much fun: tiring and potentially embarrassing. In particular, the staying in communal accommodation bit—never a laugh a minute when your digestive systems playing up (yup, been there, done that!)—and I’m afraid that yes, you won’t endear yourself to fellow walkers by being up and down all night* or ‘over-occupying’ the facilities.
Would a few compromises work? Eg commit only to a couple of weeks initially then see how it goes. (AIUI you are past peak season for the Camino now, so it shouldn’t be such an issue getting accommodation at short notice.) If you can afford it, perhaps do some of your stops as hotel/pension rather than hostel—from the perspective of better privacy, but also things like you can luxuriate in your own bath/shower. Use motorised transport to skip some of the less interesting bits.
My perspective: Pre-DC, I did quite a lot of multi-week trekking, in fairly ‘wild’ surroundings (high Pyrenees, Iceland etc.). Stayed everywhere from a bivi bag to 3* hotel, via mountain huts and hostels. Those days were long gone by the time I got pregnant (and pregnancy wasn’t straightforward to achieve), but I kept reasonably active: cycling all the way through, (gentle) canoeing at 15 and 28 weeks, and a 2-day walk/bivi at 34 weeks. Admittedly, I didn’t have morning sickness, though (a blessing of being on steroids!), nor excessive 1st trimester tiredness.
*Tim Moore talks about this in “Spanish Steps”, his account of doing the Camino with a donkey. Worth a read to help put you in the mood, if you decide to go—or as a consolatory laugh, if you don’t!