I've had this problem for a couple of years. I've tested my blood glucose hundreds of times, at all times of the day and night, and with different, but consistent, meters. The average reading is around 100. Nevertheless, my HA1c over this period, measured on a dozen or more occasions, has been around 6.0. There's no question in my mind that the two measures are inconsistent. The question for me is, which is a better guide for my behavior and health?
IMO, neither! They essentially measure the same thing, each in their own sloppy way. But is this really the most important measure?
Once there are no more blatantly high (of low, I guess) BG readings, then the priority should be to minimize insulin. This means minimizing the amount of glucose processed. In many cases and at many stages the two goals are more or less the same: reduce carbs and protein in the diet to restrict incoming glucose.
But, once BG is in the "normal" range - and I think we still need a lot of work and data to determine exactly what that really is in this context - then a better primary focus would be total insulin/total glucose. At some points, this not the same as focusing on just reducing circulating (excess) glucose and may require different actions.
For example, many people find they can get a single troublesome number down by eating a few additional carbs, e.g. dawn numbers. Of course, this causes an insulin response bringing circulating glucose down. Now, a nice happy AM number. HOWEVER, if they were getting anything under 120 or even 140 - numbers probably insignificant and not sources of any harm in and of themselves - was this really a good idea? I don't think so.
Unfortunately, we don't have home insulin meters or any way other than counting foods to estimate total glucose throughput but with whatever means we have, I think this should be the prime focus for optimum health.
I don't believe a 6.0 is anything at all to be concerned about. If I had a choice between 6.0 processing less glucose and a 5.0 processing more glucose, I'd take the former in a heartbeat. See the recent post about "B cells" (not to be confused with "beta cells"). It really supports this theory.