Corporate Captain reported a sudden RA from a non transponder twin turboprop; resulted in a NMAC.

Narrative

Flight was from ZZZ to SSI. IFR flightplan. SSI is uncontrolled and often very busy. I have been in and out of SSI for the last 20 years and have had a few issues with airplanes not reporting position; not listening to the frequency; announcing they are taking off Runway 34 when I just announced that I was rolling on Runway 04; not able to get clearance on the ground most times. So I am always on high alert around SSI. North Florida and departure area were VMC. We were talking with JAX Center (XXX.XX) who asked if we could see the airport and the pilot not flying (PNF) reported that we had SSI in sight (we were approximately 10 miles near VIXRY. We told JAX we could cancel IFR but requested to remain in flight following (until we were sure we were clear of all possible conflicts). There were small airplanes reporting position. Someone reported having us insight and would follow us for Runway 04. We heard a call Aircraft Y taking off Runway 04. There was nothing on TCAS between us and the airport so called back JAX and cancelled Radar service just outside of YATGY. Now we are off 2 frequencies and just on CTAF XXX.YY. VMC; TCAS is clear; at approximately 4 miles from the threshold of Runway 04 we received an RA (I think it was 'Monitor vertical speed.' then 'level off'. We both remember hearing level off. TCAS showed a target below us. I hit the trim; turning off the Autopilot; leveled off; looked outside to see an Aircraft Y below me at my 11 o'clock with the smallest cross section/head on view one could have. I saw no side of the fuselage; just leading edges of the wing; bulge for engines and front of the fuselage. The Aircraft Y made a roll to his right (my left). The PNF in the right seat said it was at least 45 degree bank or more when he saw the Aircraft Y last. It passed 300 ft. (or less) underneath; to our one o'clock and was gone. No radio call after take off; no transponder; climbing at a high Angle of Attack (AOA). My presumption is he took off; kept a low AOA to build speed and then pulled up crossing the approach center line at about 900-1100 ft.We were fully configured before YATGY and down to 132-135 kts. I have no idea how fast he was but it was over quickly. We never saw him until the TCAS alerted us. I saw 300 ft. on the TCAS and he was very close to our right wing when he went by. We had landing lights on; pulsating lights; Nav lights; strobe lights. We were as slow as we could get and looking specifically for TCAS targets and wild targets like this one.I received my Private pilot license in Year XXXX. I seen a lot; learned a lot; but I have never been this close to dying in an airplane before. This was the closest; most dangerous; and unprofessional handling of an airplane I have encounter in many years of participating in aviation.Recommendations: 1. everyone should follow the rules and when rules are not enough use good judgement and common sense.2. I really wish the FAA would establish a Tower at SSI even for 10-12 hours per day or during the busy season.3. I'm baffled that there was no transponder. The cross section could not have been smaller. I would guess his AOA to be 30-45 degrees to be that close and straight on from beneath us. I guess this goes back to No. 1 following the rules. Turn on your transponder. SSI is crazy enough if everyone is going slow and on the lookout. This pilot obviously didn't have those same concerns.