I know this is six weeks old and we don't generally respond to old posts, but I didn't see this one initially. This is exactly what I do for a living; if the OP worked for my employer I'd be the one he'd be talking to about this. For benefit of future readers I want to add an explanation of what happened here and why.
Since this is employer based coverage no, you do not have a leg to stand on. The coverage is retroactive to May 1 and if you had incurred any eligible bills during the month of May, they WOULD have been paid. At worst, you might have had to pay up front and be reimbursed after the fact. But the coverage was effective May 1, even if BCBS did not receive the application until May 20th.
How do I know this for sure? Because that's the law. IRS regulations require that all employer sponsored coverage that is not elected during Open Enrollment, be effective the actual date of whatever the Qualifying Event is. That might be a new hire, the birth of a baby, a marriage, loss (or gain) of other coverage, or some others. I know it wasn't an Open Enrollment (though if it were, the ONLY possible date would be the first day of the plan year) because the election was made after the effective date. That does not happen during an Open Enrollment, but it DOES happen during an IRS approved qualifying event. In that case, there is a 30 (occasionally in very rare circumstances 60, but mostly 30) day window to make elections, and that window STARTS on the date of the qualifying event. If the QE was May 1, then as long as the elections were made between May 1 and May 30, the effective date of coverage is May 1. That is the law. There is no other answer.
And since the coverage effective date was May 1 (even though retroactively) then the OP owes the premium for the month of may. Making coverage effective June 1 was NOT a legal option.