How to run a flex duct branch through a unreachable trunk?
Some options that come to mind are using a low profile drill to start a hole and then awkwardly cutting it with tin snips or a nibbler if you have enough space.  With a little more space you may even be able to fit a hole saw with a low profile drill.  Some &quot;Adjustable&quot; hole saws have shorter length than common arbor and cups.
If you really can't fit anything in there, from your picture it looks like you have access to the opposite side, so you can get a 5&quot; holesaw and a 18-24&quot; hole saw extension bit and drill through both sides of the duct.  Once you're done, seal the hole you don't want with a metal plate, self tapping screws and that grey duct basher goo.  That gets you a hole and just leaves you with needing to screw your flange on, which could still be a huge pain, but again a low profile drill should do the job with only 5-6&quot; clearance.  If you don't have 5-6&quot; clearance, you don't have room for your duct anyway, so I assume you have at least that to work with.  There may be some adhesive that is considered acceptable, but I've never seen a professional HVAC guy forego screws.
Hex head duct screws are particularly easy to work with with a magnetic bit, but an option for really hard cases is to drill a pilot hole and then put the screw in by hand with a stubby screwdriver.