First, thanks for taking on the little guy!
If you are feeding BML, you should not be giving extra calcium supplements. You can give him a little bit of the chicken/turkey as a treat, but its not needed as part of the diet and will throw off its balance.
Also, BML has specific fruit/veggie lists - veggies should be mixed veggies of corn, green beans, carrots, and peas. Veggies should include melons, berries, apples, etc.
I would try sticking to the original diet plan to see if that resolves the constipation and if it does not, get him to the vet to be checked out for medical causes.
Are you planning on getting him a friend soon?