Shared custody - I claim my child on taxes as their guardian does not work.
You need to make sure you qualify to claim the child as a dependent. The fact the guardian does not work does not factor into that. If the child does not live with you for at least 1/2 of the year (183 nights), you might not be able to claim the child.
https://www.irs.gov/pub/irs-pdf/p501.pdf#page=11
Next year, I will need to
file my 1095-A, do I need to add my dependent to my household count even
if I don't provide their insurance?
Yes (assuming your claim the child as your dependent). The program will automatically do that.
Do I need to purchase insurance for my dependent even if they already have it?
No. But you should notify the Marketplace that you will have a Household Size of two, so they can adjust any Advance credit accordingly.
Do I withdraw from my insurance? Or do I just not claim my dependent on taxes?
There is nothing wrong your situation and nothing wrong for claiming your dependent (IF the child qualifies as your dependent). Just contact the Marketplace to tell them about your correct Household Size.