Does anyone have any recommendations on how to handle a leave denial?

Your 1SG isn't exactly wrong. Most units get opportunities for leave twice a year. While it certainly makes sense to go see your family, what is the issue with waiting until the next block leave/opportunity leave like everyone else? If your commander grants you an exception when your situation isn't exceptional, then they have to grant an exception everyone or else it's favoritism.
If it's so important you can't wait four more months, explain to your 1SG that you would be taking this leave in lieu of the opportunity leave in the summer. Otherwise, all your asking your 1SG is for an exception for a very routine occurrence (birth of a dependent).

You've got a few different paths you could take here. A lot of it depends on how many bridges you can afford to burn....

First thing that I looked at is the fact you are no where near use/lose status for accrued leave. As a Section Chief, I would most likely put you at the bottom of the priority list for troops requesting leave time, especially since you just returned from a leave period.

Second, you just returned from a leave status. Makes it more difficult to justify granting another block especially since you are asking for nearly the ENTIRE month of February (as pointed out by MSG (Join to see)). Just because YOU are not aware of any upcoming FTX, training, exercises, or other unit events does not mean there isn't something coming down the pipe. Some things are "close hold" until just before execution. Your CO, CSM, and 1SG are under no obligation to reveal that information premature as justification to deny leave. It sucks but many units have block leave restrictions for a reason. Even the AF had times you could or could not take extended leave periods. I hated it also but such is the fact of military service. If it helps any, I still didn't like the policy even as a SNCO but I had to (and did) abide by it.

Third, you could submit a leave form (I think the Army uses those like the AF did) in hopes of forcing the issue. There should be a block on the form where your CoC can annotate either approve or disapproval of the request. I'm betting as soon as your Platoon/Company leadership learns of the 1SG's denial, they will most likely be unwilling to sign as approving. Assuming the request is disapproved, use the form for later justification to regain any lost leave days in the event of a use/lose situation later on. WARNING: This tact will most likely sour the relationship you have with your CoC. Your entire supervisory chain will have to go on record as being against granting leave time. Putting your leadership in that situation is never a good thing.

Fourth, you could utilize your Commander's Open Door policy and speak with them directly. You'd have the opportunity to plead your case and hope to sway the CO to your side. You'll want to make sure your entire CoC is aware of the situation before you invoke that option. Letting your Platoon SGT or Company leadership get blindsided is a surefire way to make your life infinitely less pleasurable moving forward. WARNING: This tact will also most likely put you on the 1SG's s#*t list as the 1SG has already expressed disapproval and given valid justification for said denial. Any 1SG worth the rank will have all sorts of creative ways to make your life a living hell without giving the appearance of retaliation or harassment.

Also, keep in mind, unless the rules have changed from when I had a newborn overseas many airlines have a minimum age restriction (ie: baby must be at least two-three months old to travel). That may have changed (he is 20yrs old as of this posting). My infant son had to be six months of age before the airlines would allow him to travel. Both civilian and military aircraft had the restriction.

1. You can't handle it, you have to cope with it. 2. It is in the hands of your CO. 3. You just came off leave for crying out loud, and now you want another month? Just when are the other folks in your unit suppose to go on leave? 4. Think your child is too young to travel CONUS and probably need some shots. 5. If you aren't in the take or lose bucket, then you have no gripe. 6. I imagine everyone in the unit wants to go home, you had your leave now let someone else go. And finally, have you checked to see how many troops in your unit didn't get the chance to see their newborn till after a deployment? Sorry, PFC but the Mission always comes first, people second; either realize and live with or ETS out. Congrats on the baby.