Is it bad to not claim a child as a dependent?

Child's mother won't let you? That's not an option. The child is your dependent, and youre responsible for it. You take that birth certificate to DEERS and enroll the child. Then you enroll your child in tricare and tricare dental. Whether you'll get in trouble should be the least of your worries, you need to provide medical coverage for your child. The only time this doesn't apply is if the child's mother already has them enrolled in DEERS under someone else.

So the childs mother has no say in letting you put your child into DEERS. If you are on that childs BC then it is your right to put that child into DEERS she cannot tell you no. Now if you don't have an official certified BC and SSN then you wont be able to and then that is where the COURTS will come into effect. But that is absurd especially with medical ect for your child. Get a lawyer and fight for your Child's rights.

There is a risk to you, and may be theoretically risk to the child if it isn't covered by health insurance.
"Assumptions" based on your post:
1. She is petty (your words, not mine).
2. Your name was put on the birth certificate (BC), and you did not challenge your name on the BC within any specific time period that may be required by your state, so paternity is acknowledged/established.
3. Were something to happen to you on active duty, there may be benefits that the child could be eligible for, but would have a more difficult time obtaining if not already in DEERS.

In addition to speaking with JAG, consider reading AR 600-99, chapter 2, and appendix B.

You can Google how to obtain a copy of a birth certificate for the state in which the child was born, and obtain a certified copy (normally raised seal). You will require an official copy of the BC for most any action with any government agency, including the military.

Once you obtain a copy of the BC, you can visit a Social Security office for the documents required to obtain the SSN of your child.

Petty people do petty things. If you provide not support, she could wait for a couple of years, and then file a claim for back child support. That would likely mean trouble for you, and it would be a financial liability.

You should Google child support requirements for a father on the BC without a court order for support in the state in which the child resides. Even without a court order, since you didn't oppose your name on the BC, you may have met the legal requirement for that state, and therefore, the military, of paternity. You can Google what the time limits to oppose your name on the BC for that state.

Medical care isn't cheap, and IMO, everyone can benefit from regular checkups. Additionally, life happens, and sometimes people need emergency medical care. Medical care is very expensive. You may (or may not) end up with a financial liability, especially if an emergency arises. I'm not an attorney, and that's who can best explain such a risk, if any.

AR 600-99 requires Soldiers to provide support for your child, but my reading of AR 600-99 leads me to believe that there is not action, unless there is an inquiry (by the child's mother).

This example in appendix B seems to fit your circumstances:
B-1. Example 2. Assume the same facts as in example 1, except that the acknowledgement of paternity consisted of his agreement to being named as the father on the birth certificate when the child was born a year ago. The father has not raised any subsequent challenge to the validity of the birth certificate. The state law where the birth occurred will determine resolution of the support issue. If, as in many states, agreeing to being listed as the father and not challenging the document within a specified number of days is the judicial equivalent of a court order establishing paternity, then there would be a legal order establishing paternity pursuant to paragraph 2–2b. The soldier would have an obligation to provide support to that dependent child. Since there is no court order establishing the amount of the child support, the soldier will provide the appropriate support required under paragraph 2–6.

I didn't see an example in 2-6 that fits your scenario, but that is a question for SJA to answer for you.

I encourage you to obtain a *certified* copy of the BC and keep it with important documents in a fireproof/resistant bag at a minimum, obtain the SSN, and then enroll the child in DEERS. I also encourage you to seek SJA's advice. Note that the Court Martial Convening Authority in your chain of command can give you an exception to policy with regard to financial support, which may provide you protection from pettiness.