CPR X REFUSED ATC INSTRUCTIONS CLRNC BECAUSE OF WX TSTMS.

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CPR X DEPARTING RAC WITH AN UNORGANIZED AREA OF TSTMS OUT TO THE ENE OVER LAKE MICHIGAN. BEFORE TAKING OFF; WE TURNED OUR ACFT TO THE E AND TURNED ON THE RADAR. WE DETERMINED FROM BOTH VISUAL CUES AND RADAR THAT APPROX E HDG AND 3000 FT MSL WOULD WORK. ANYTHING S OF E WOULD ALSO WORK; HOWEVER; MKE IS USUALLY UNABLE TO DO THIS BECAUSE OF ORD APCH CTL AIRSPACE AND ORD CTR RESTRS FOR HDOFS. WE WERE ISSUED DEP CLRNC FROM RAC (VIA REMOTE) HDG 090 DEGS AND 3000 FT MSL. ONCE AIRBORNE WE REALIZED THAT 090 DEGS WOULD NOT WORK. TSTM CELLS AND LIGHTNING AT 12 O'CLOCK; AND INFORMED ATC WE WOULD REQUIRE 30 DEG R (120 DEGS). ANY L TURN WOULD NOT WORK. ATC DIDN'T RESPOND FAVORABLY AND WE TURNED ANYWAY TO AVOID A CELL AT 12 O'CLOCK AND 3 NM. ATC WENT ALONG WITH IT THEN AND ISSUED A CLB TO 7000 FT MSL. WE COMPLIED. ATC ISSUED A L TURN -- WE REFUSED. CTLR SAID ORD CAN'T HAVE YOU THERE; TURN L HDG 020 DEGS; WE AGAIN REFUSED. CTLR WAS BECOMING AGITATED; INSTRUCTED US TO TURN R TO 320 DEGS; WE COMPLIED. UPON REACHING 320 DEGS WE TOLD ATC WE WOULD FLY THAT HDG FOR 5 NM TOPS. ATC TURNED US TO 360 DEGS; WE TURNED AND TOLD ATC WE HAD TO TURN WITHIN 2 NM. ATC SAID OTHER ACFT OFF OF MKE WEREN'T HAVING PROBS GOING THROUGH THAT AREA (IN AN INTIMIDATING TONE). I SAID I DON'T KNOW ABOUT OTHER ACFT; BUT WE HAVE TSTM CELLS AND SEE LIGHTENING. IF IT'S EASIER FOR CTR; 3000 MSL WOULD BE FINE -- ANY ALT. SHE SAID THAT'S NOT CTR'S AIRSPACE AND THAT WON'T HELP ME (DURING THIS A LARGE CLOUD TO GND (WATER ) LIGHTENING BOLT STRUCK AT 12 O'CLOCK ALONG WITH MUCH CLOUD TO CLOUD LIGHTENING). SHE ASSIGNED 070 DEGS -- WE WERE ALREADY THERE; AND HANDED US OFF TO CTR WHICH WAS VERY ACCOMMODATING. SHE WAS BUSY. SUPPLEMENTAL INFO FROM ACN 270977: AFTER TKOF; WE OBSERVED MORE WX E OF OUR POS AND A CLR AREA TO OUR S; SO ON INITIAL CONTACT WITH MILWAUKEE DEP; I INSTRUCTED THE CO-CAPT TO REQUEST A 180 DEG HDG FOR ABOUT 5 MI BEFORE TURNING E. MKE AUTH THE HDG. ABOUT THE TIME WE WERE READY TO TURN E; THE CTLR CALLED US AND ASKED HOW MUCH FURTHER S WE WERE GOING TO GO. WE INFORMED HER WE WERE TURNING E; AND SHE APPROVED THE HDG. SHORTLY THEREAFTER; THE CTLR ISSUED A CLRNC TO TURN L TO 360 DEG AND CLB TO 7000 FT. I SAID 'UNABLE' AND THE CO-CAPT RELAYED THIS TO THE CTLR. THE CTLR SAID 'CTR DOESN'T WANT YOU IN THAT AIRSPACE.' WE SAID WE COULD STAY AT 3000 FT AND CONTINUE E. THE CTLR THEN SAID; IN A SHARP TONE;' CTR DOESN'T OWN THAT AIRSPACE AND I DON'T HAVE TIME' (OR WORDS TO THE EFFECT). THEN SHE SAID 'NO ONE ELSE IS HAVING ANY TROUBLE GOING THROUGH THERE.' THE CO- CAPT THEN RESPONDED; 'I DON'T KNOW ABOUT OTHER ACFT; BUT WE HAVE CELLS AND WE SEE LIGHTENING!' THE CTLR'S TONE WAS VERY UPSET. SHE SAID 'MAKE A R TURN; NOW; TO 330 DEGS;' WE COMPLIED. UPON ESTABLISHING THE 330 DEGS HDG; WE WERE VERY CLOSE TO THE WX AND INFORMED THE CTLR THAT WE WERE TURNING R TO 300 DEGS. SHE ACKNOWLEDGED. WHILE ON THE 300 HDG HDG; LARGE BOLTS OF CLOUD-TO-CLOUD AND CLOUD-TO- SURFACE LIGHTENING WERE OBSERVED VISUALLY. WE INFORMED THE CTLR THAT WE WERE TURNING TO A 020 DEG HDG. AGAIN; SHE ACKNOWLEDGED. ONCE MORE WE INFORMED CTLR THAT WE WERE TURNING R TO 070 DEGS. HER ONLY RESPONSE THIS TIME WAS 'CONTACT ZAU ON 125.1;' WHICH WE DID. CTR WAS MOST HELPFUL AND PROFESSIONAL AND WITHIN A COUPLE OF XMISSIONS; WE WERE CLR OF THE WX AND THE REST OF THE FLT WAS UNEVENTFUL.