The antibodies to wheat and gluten are not the same antibodies that occur in Hashimoto's. So you can have difficulties with wheat and gluten, but not have Hashimoto's (or vice versa), or you could have both.
However, if you have one autoimmune condition your chances of developing another one are higher than they would be for someone who didn't have any autoimmune problems at all.
As for getting a TSH level high enough to persuade your doctor to treat, that's a difficult one.
You might want to find out for certain if you have Hashimoto's. Take a look at this page, particularly the table lower down the page :
Damping down antibodies (if you have them) will lower your thyroid hormone levels and raise your TSH. For info on this see these links :
The author gets good reviews and is a sufferer herself.
The only other thing you can do is to follow the suggestions I made earlier - fast (except for water) before your test, and get the blood taken as early in the morning as possible.
If you have pituitary damage you might never be able to get your TSH up to a high enough level to get treated. In that case you will have to hope your Free T4 and Free T3 drop below the range. But you could be waiting a very long time. The only other alternative is to treat yourself.