Ok. I'm getting on my stump, again. TxScout,
Your son does not need his SM's permission to start any MB. He can have as many partials going at one time as he wants. Period. Your son is, however, obligated to seek the advise* of the SM (or his/her designee, which must be an ASM) prior to starting on a MB. This is the opportunity for the SM to counsel, advise, guide the Scout; but, if the Scout insists, he must always* be given the blue card upon request. This cannot be refused, unless mom or dad says so. The SM (or designee, an ASM) does have the right to assign which MBC your son must use to work on the MB. Other than safety, the SM has no say. If the SM does not specify which MBC to use, then your son has the right to choose any district- (or council-level, as the case may be) certified MBC he wants.
The reason for getting the blue card is not to show the SMs "permission" was granted to take the MB; the reason is to verify which set of requirements were the "current" requirements at the time the MB was started... which can become a critical factor down the road. (See other related posts.)
That being said, wagionvigil's advise concerning "Is this a battle worth picking?" is good advise. Your son's SM is (most likely) doing business the way he was taught... it's exactly the same way I was taught. But, it's wrong.
(*) IMHO, the only thing a SM should be thinking when considering saying "No" to a Scout who asks for a blue card is, "Do you have your parents permission to take this MB, at this time?" Scouting is not w/o risks. Common sense dictates that moms, dads, and SMs, should be in the business of looking out for Scouts' safety. Otherwise, like I've ranted about before, let him take as many as he wants.