I am with you Dh here. Two sit-ups, especially at the speed or pilates, shouldn't trigger such a reaction.
Do you feel nauseous, dizzy? When you say "sick" what do you mean?
We could give you a list of abs that do not involve crunch or sit-ups but I am not comfortable here, because you need to find out why.
Is it only the crushing together the muscles in the typical crunch/sit-up / C-shape movement or also the compressive type (suck all air in as in pull belly in for photo, then keeping the vacuum, breathing in and out? , what about side. Standing up tall, arms towards ceiling and slowly bending the waist keeping arms up? What about hanging from something and lifting your knees towards our chest?
When you go to the GP, don't say " I can't do abs because I feel sick", but say whenever I bend my torso, especially in the abs section of a workout, I feel sick for hours.
Say that if you are in a class in a gym, you have to leave the class, because you are too sick to continue.
Say it is impacting your ability to exercise. Your core works with your skeleton to keep you up straights. One needs a good core.
I would ask for some sort of imaging because you might have a bend in the bowel or something similar totally innocent .