ZKC CTLR CHALLENGES B737 FLC AS 'OFF COURSE.' FLC TAKES NO ACTION; STATING THEIR EQUIP SHOWS THEM 'ON COURSE.'

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DEPARTED STL RWY 30L. PRIOR TO 500 FT AGL; TWR GAVE US NEW HDG OF 210 DEGS. CONTACTED DEP. THIS HDG PUT US E OF THE STL 214 DEG RADIAL OF THE DEP. MULTIPLE ALT CHANGES. CONTACTED ZKC. HE GAVE US 230 DEGS TO JOIN LINDY 2 MAPLES TRANSITION. I PROGRAMMED LNAV TO INTERCEPT DEP. CAPT WAS BACKING UP DEP. ON STL VOR (17.4 214 DEG RADIAL) LNAV CTRED AND VOR CTRED ACFT JOINED. ABSOLUTELY NO OVERSHOOT W OF COURSE. GIVEN FREQ CHANGE TO ZKC 125.5. SHE IMMEDIATELY TOLD US TO TURN 15 DEGS L TO JOIN THE RADIAL BECAUSE WE WERE TOO CLOSE TO COMPANY TFC. SHE SAID WE WERE 5 MI W OF COURSE. WE ADVISED HER WE WERE ON COURSE AS INDICATED BY BOTH VOR AND LNAV. SHE AGGRESSIVELY DISAGREED WITH US. A FEW MINS PASSED WHEREBY OUR HDG AND COURSE NEVER CHANGED FROM OUR INITIAL HDG WE JOINED BY 215 DEGS. SHE THEN ADVISED US WE WERE NOW ON COURSE AND GAVE A FREQ CHANGE. CAPT THEN ADVISED HER 'FOR THE RECORD; OUR HDG OR COURSE HAS NOT CHANGED 1 DEG SINCE THE INITIAL INTERCEPT PRIOR TO YOUR FREQ.' (SOMEHOW WITHOUT A HDG CHANGE WE MAGICALLY CORRECTED 5 MI W OF COURSE TO RIGHT ON.) SHE REPLIED 'FOR THE RECORD; OUR HDG HAD CHANGED AND YOU ARE BACK ON COURSE.' VERY ADAMANT OF THE FACT WE WERE OFF COURSE AND NOW CORRECTED. THIS DID NOT HAPPEN. OUR HDG NEVER CHANGED AND WE NEVER OVERSHOT THE DEP RADIAL. SHE MANUFACTURED THE EVENT FOR SOME REASON UNKNOWN TO US. WE NEVER SAW TFC ON TCASII AND SWITCHED FREQ WITHOUT ANY FURTHER DISCUSSION. I WOULD LIKE TO KNOW THE REASON THIS DISCUSSION TOOK PLACE SINCE WE WERE NEVER W OF COURSE. SUPPLEMENTAL INFO FROM ACN 523447: I AM PERPLEXED AS TO THE CTLR ADVISING US THAT WE WERE 5 MI W OF COURSE WHEN WE SHOWED NORMAL NAV ON COURSE ON THE LINDY 2; YET A FEW MINS LATER THE CTLR ADVISES WE WERE NOW ON COURSE. OUR HDG AND TRACK NEVER CHANGED DURING THE ALLEGED OFF COURSE DEV. WHEN CTLRS USE THE TERM (FOR THE RECORD) IT MAKES PLTS VERY ALERT. FROM TIME-TO-TIME; WE OVERSHOOT RADIALS; BUT THIS WAS NOT THE CASE. WE HAD 2 SEPARATE NAV SYS OPERATING AND BOTH CONFIRMED OUR POS ON THE DEP.