Based on the information you provided I do not believe the unit can stop you from attending BLC.
If you are not flagged or otherwise inedible for BLC (example on profile, fail to meet any standard required for BLC). You cannot be stopped from attending BLC.
However, they may be concerned that if CPS takes your children you would be recalled from BLC wasting a slot. If this is the case, I believe the decision may be appropriate.
Things to Consider
1. Use your chain of command up to and including the CSM and Battalion Commander. If they give you an answer that clearly follows a regulation then you have your answer.
2. Should you not be satisfied you can contact the IG and/or JAG for assistance.
3. If you are not allowed to go to BLC and the decision is not based on regulation or fact you can consider filing an article 138 against your commander. Here is a link that explains the Article 138: http://asktop.net/articles/25189/
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This response is based on the information you provide. My comments do not represent the US Army or US government positions. Furthermore my comments should be used for information purposes only.