If the thyroid is adequately treated, it's likely not playing a role in the constipation. Would consider a GI consultation.
You could also try a magnesium tablet daily and extra vitamin c.
You don't mention your thyroid levels, are they in range?
I had dreadful constipation when I was on thyroxine (synthetic T4.....12 years of it) which resolved itself 6 months ago when I switched to Armour a whole thyroid med containing T1, T2,T3,T4 and calcitonin
Go here
www.stopthethyroidmadness.com
and here for Armour info
www.armourthyroid.com
Do you have any labs and ranges you can post?