Gluten Free and Matzat Mitzvah Rabbanit Debbie Zimmerman

Your situation is definitely a difficult one, to answer it we will have to ask a bigger question: is there any obligation to perform a mitzvah if it will make one sick? In general, this question is divided into two parts:
1. What is the status of the mitzvah discussed (in this case matzah)?
2. How sick will the individual get (i.e. is it life threatening, minor ailments – or something in between)?
1. The Mitzvah of Matzah
We will begin by understanding the mitzvah. The mitzvah of eating matzah on the first night of Pesach is considered a positive Torah commandment (מצות עשה דאוריתא) (TB Pesachim 120a,). This is fulfilled by eating one kazayit of matzah on the seder night. Rabbinic stringencies and traditions add an additional 4 kazaytim of matzah, for a total of 5, but the consensus is that one is exempt from positive rabbinic obligations if it makes them sick (Shulchan Aruch/Rama YD 155:3). Therefore, the discussion here will focus on the 1 kazayit of matzah seder night.
While the Rama rules that matzah should ideally be made from wheat, both he and the Shulchan Aruch agree that any of the 5 grains can be used to make kosher matzot (OC 463:1). Recently some scientific and archaeological evidence has been brought that claims that the oats we eat today are not the same grain described in the mishnah, however this opinion has been rejected by many major poskim (such as Rav Moshe Feinstein, Rav Elyashiv, and Rav Y.D. Soloveichik, Rav Ovadiah Yosef, Tzitz Eliezer, Rav Shternuch).
Therefore, while a person without any complications should preferably eat wheat matzah, it seems clear that an individual with health problem can opt for matzah made of oats.
2. Performing a mitzvah that will make one ill
From my understanding the medical community is still researching the various causes and effects of non-celiac gluten intolerance. Without any more details, and without knowing the underlying cause of your symptoms the halachic discussion will have to address your symptoms based on an assumption that any complications you have are not life-threatening.
Classic halachic texts that discuss illnesses that interfere with mitzvah observance generally divide into categories based on severity – whether or not the illness is life-threatening, and how ill does the person feel. From what you describe it seems that when you eat gluten you fall into the category of a non-life-threatening illness (חולה שאין בו סכנה), and when you eat oats you are in the category below that, מיחוש, which refers to discomfort without incapacitation.
To clarify, if there is even a chance that eating matzah is life threatening such a person is exempt from eating matzah, and it is quite possibly a sin for them to eat it (מצוה הבאה בעבירה). But the status of a חולה שאין בו סכנה is not as clear. The Maharm Schik rules that as long as symptoms do not appear immediately such a person is obligated to eat matzah, even if they become incapacitated by their illness (OC 120:1). Other poskim (halachic decisors) disagree. The Chida (ברכי יוסף) cites an explicit mishnah that exempts a sick person from the mitzvah of succah and expands it to exempt any חולה שאין בו סכנה from positive Torah mitzvot (Birkei Yosef OC 640:5). Similarly, Rav Shlomo of Vilna explains that the Torah does not require people to make themselves sick to perform mitzvot, and therefore holds a similar exemption (Binyan Shlomo OC 47).
A few contemporary poskim discuss celiac disease, but don’t explicitly mention non-celiac gluten intolerance. The Tzitz Eliezer (19;22) cites Rav Shlomo of Vilna and rules that there is no obligation for a person to make themselves sick to fulfill a mitzvah, even the mitzvah of matzah. Nevertheless, he rules that if one desires they can also choose to fulfill the mitzvah of matzah and recite the blessing – and this is not considered a מצוה הבאה בעבירה.
Similarly, Rav Asher Weiss (Minchat Asher III 42-43) rules that one is not only exempt, but also prohibited from fulfilling Torah commandments that will cause irreversible damage to life or limb. In the case of a חולה שאין בו סכנה the individual is exempt but may choose to fulfill the commandment. And in cases of pain or discomfort the individual should push themselves to fulfill the commandment.
Additionally, the Shulchan Aruch rules that when one eats a single kazayit of matzah they should wait to eat it for afikomen (i.e. begin their meal with marror, wait to wash until the end of the meal, and at that point recite the blessings of motzi and matzah); this way they fulfill both the mitzvah of matzah and that of afikomen.