B737 FLT CREW DURING APCH TO A NON RADAR; HIGH TERRAIN ARPT; ENCOUNTERS SEVERE WX AND ELECTS TO EXECUTE A MISSED APCH PRIOR TO THE MISSED APCH POINT. ATC WAS UNABLE TO PROVIDE TERRAIN CLRNC AND DIRECTS FLT CREW TO PROVIDE OWN TERRAIN CLRNC.

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ON APCH TO ZZZ1 VIA THE RNP RWY 11; WE RECEIVED A PIREP FROM A LEARJET 5 MINS AHEAD OF US THAT THERE WAS LIGHTNING; TURB AND CONVECTIVE ACTIVITY ON FINAL TO RWY 11. AS WE APCHED; WE BEGAN TO EXPERIENCE HVY RAIN AND NOTICED RADAR ECHOES ON THE APCH COURSE. WE MADE THE DECISION TO MISS THE APCH AND REQUESTED CLRNC DIRECTLY TO ZZZ2. ATC DENIED OUR CLRNC AND ASKED US TO FLY THE PUBLISHED GND TRACK OF THE APCH. SINCE THAT WOULD TAKE US RIGHT INTO THE WX WE WERE TRYING TO AVOID; I EXERCISED MY EMER AUTH TO PROCEED DIRECTLY TO ZZZ2; REQUESTED AND CLBED TO 7000 FT MSL (THE MSA IS 6400 FT). ATC ADVISED US THAT WE WOULD BE RESPONSIBLE FOR OUR OWN TERRAIN SEPARATION UNTIL REACHING ZZZ2. AT ZZZ2 WE HELD AS PUBLISHED ON THE RNP 11 APCH AND AFTER HOLDING AND LETTING THE WX CLR; WE PROCEEDED INBOUND FOR LNDG. PREVENTIVE MEASURES: THE DIFFICULTY IN A NON RADAR ENVIRONMENT IS WHEN A DEV AROUND WX IS NECESSARY; ATC CAN'T HELP AND THE PLT DOESN'T HAVE MUCH TIME TO DECIDE WHAT TO DO. NOT ONLY IS THE WX A FACTOR; BUT THE TERRAIN IN THE AREA ADDS THE EXTRA LAYER OF DIFFICULTY. THE TRACK TO ZZZ2 IS ENTIRELY OVER WATER AND HAVING THE GPS GUIDANCE AVAILABLE; I BELIEVED THAT OUR ACTION WAS THE BEST CHOICE. ATC'S RESPONSE TO OUR REQUEST WAS UNDERSTANDABLE BUT AT THE SAME TIME; WE WERE BUSY FLYING THE ACFT AND MODIFYING OUR NAVAIDS. THIS WAS NOT THE TIME TO TELL US WHAT ATC COULDN'T DO. PERHAPS THE WAY TO HANDLE THIS IS TO HAVE ATC RESPOND IS CLR THE ACFT IS POSSIBLE TO THE MSA (OR HIGHER) AND TO THE NEAREST IAF; WITH THE PLT BEING RESPONSIBLE FOR TERRAIN CLRNC.