What advice or tips do you have on how to manage attending BOLC with an infant?

I was going to suggest that you option out of BOLC until they are 1-year old (you should be given an automatic six-month extension on the waiver due to the 365-day pregnancy deferment* if you so chose), but my SWAG from your statement is that you're already at the 3+ year mark from being commissioned. However, if you do want to go this route, then discuss it will your leadership and HRC about extending the waiver for another six months (or thereabouts). It would give you some more time to make arrangements prior to BOLC.

If you don't go that route, there are a number of Child Development Centers* (CDC) on Fort Gregg-Adams that you would be eligible for as you'll be on active duty orders when you're TDY there. Another option would be to contact the Family Child Care (FCC) providers on post (there are four listed for Ft. Gregg-Adams).

If you don't know what the CDCs or FCCs are, there's a link below* with the descriptions of the programs, requirements, standards, oversight, etc. However you can think of CDCs as more of the traditional "child care facilities" while the FCCs are usually military spouses that have been accreditated by DoD and provide care at their homes.

There's usually a waiting list, but you can be added by creating an account at militarychildcare.com*
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* Army Pregnancy Policy (2022-06) - https://armypubs.army.mil/epubs/DR_pubs/DR_a/ARN35255-ARMY_DIR_2022-06-000-WEB-1.pdf
* Full descriptions of military-operated Child Care Programs - https://public.militarychildcare.csd.disa.mil/mcc-central/mcchome/military-operated-child-care-programs
* https://installations.militaryonesource.mil/military-installation/fort-gregg-adams/child-and-youth-services/child-care
*

I'm going to make an alternative suggestion, find a trusted relative and leave the child with them. I have no idea how difficult the Transportation BOLC, but I have completed my Basic Infantry and Engineer Advance Course. Infantry Basic is extremely busy and I was in the field a lot. I couldn't imagine trying to complete it and care for a young infant at the same time. Even cantonment in the Engineer Advance Course was usually 12 hour days.

Contact Ft Gregg Adams and see what kind of existing family services they have. Another route would be to see if there are any Facebook groups for Gregg Adams families/dependents/spouses and see if anyone has any childcare options for a 6mo old. Or if you're lucky, perhaps someone there may even runs an in-home day care that could help you out.