My 3 inch by 9 grade 316 stainless steel deck screws are breaking
For the screw that is stuck... The easiest way to deal with it is to chop it off and then to take a bigger bit and drill it down a good 1/8&quot;+ so it doesn't ever stick out.   Sure you can spend a ton of time backing it out but I wouldn't bother with it.
I will give you two general pieces of advice when doing something like a deck...

Test out your screws with the materials you have.  I certainly wouldn't trust most stainless steel screws to go in that deep without a pilot - without breaking.   Self-tapping means you don't need a pilot/starter to get it to catch easy.   It by no purpose means you don't need a pilot hole.   I would test a couple of sizes of pilot holes and see how big you can go while keeping your screw really snug.

Back your screws out halfway once you get to 3/4 of the way down.   This helps clear the threads and it gives your screws a break from the amount of heat.   I do the same technique when doing any kind of wood join with screws.   It reduced breakage and stripping the heads by 5 fold.