You need to meet the terms that BMI puts forth if you play their material. Their website deals with this issue and they CLEARLY tell you that it is the owner of the venue, not the DJ who is liable.
Yeah, and those boxes that pop up on computers CLEARLY tell you that the IRS is going arrest you if you don't call this number and give them your credit card. That doesn't make it true.
Or, for an example a little more on point with how BMI operates, police will often say something like "I'm going to need to come inside and look around". The implication is that they are legally allowed to enter simply by saying so, when this is completely untrue - they actually cannot come in at that point unless you signal your acceptance for them to enter. What happens is that 99% of people will step back and hold their door open for the officer after he says that, thus legally extending the officer an invitation to enter (even if the invitation was caused by an implied lie).
Sure the DJ may have claimed to have the licenses required, but the business owner is the one who is required to obtain the licenses.
BMI can state that the venue is responsible all they want, but if you read between the lines on what they actually say on their website, you can have DJs (or whoever) be responsible for the licensing, it's just that you may be liable if the DJ isn't properly licensed. If the terms of a contract with a DJ requires the DJ to be responsible for the music, then there's a good chance that's exactly how it will work - either by direct assignment of responsibility, or by being responsible for any damages the business incurs due to their negligence.
Simply put, BMI words their statement like this because they want to double-dip into the licensing revenue stream. Even if the DJ already has all the proper licenses to play it in whatever venue they want, BMI will use vague language about who needs the license and use scare tactics to try make venue owners pay again for licenses that have already been paid.