Essentially no unfortunately.
Any healthcare professional can order any test even if it is prohibited by the CCG, but there has to be a clinical reason for it. Not being able to pass the DVLA test wouldn't qualify, which is also of course why you have to pay for the DVLA tests.
However, I don't think any test your GP can order will help much anyway. When they do the
CDT test they also do a basic liver function test to balance the test against blocked bile ducts. For the test to give a "wrong" result after such a long time there are only usually three explanations.
1) End stage liver disease or COPD heart disease. You wouldn't need a test to tell you if that was the case.
2) You have an unusually high amount of D-transferrins in the blood. The CDT test checks for C-transferrins but about 0.01% of Caucasians have abnormally high levels of D-transferrins and this gives false positives on the test.
3) A disease called CDGS (Carbohydrate deficient glycoprotein syndrome) but that is rare, think lottery ticket rare.
Out of those (2) is obviously the most likely albeit still unlikely overall. However, even if your GP did the test and found you had elevated D-transferrins, it wouldn't help you much, because you would still need a doctor to write a letter explaining why the results make the CDT test unreliable - and again you would need to arrange that privately.