"And there are types of mines that are difficult to either detect or clear."
A few dozen nuclear warheads should sweep the channel clear, no matter how hard the mine is to sweep.
After all, being hard to detect doesn't give one the ability to withstand 1,000+ psi dynamic overpressure. :)
I was thinking the same thought. Some B83s adapted for underwater use and detonated at full yield (1.2 megatons). The resulting micro-tsunami may even wipe out some coastal land batteries as well.