I've constructed many a sub enclosure from 19mm MDF in my youth and my experience has been this: 1) Any cross-linking PVA will be more than adequate, and 2) making MDF air-tight is harder than it might appear.
Before I started working with Titebond I used some basic yellow glue with MDF and found that the MDF would fail long before the glue joint ever would. For the laminated plates yellow glue is fine, but for filling gaps you'll need epoxy.
I'm not familiar with how a Sonsub works exactly, but if you're going to be dealing with very large air pressures inside you may want to consider a sealer for the inside and silicone for all the joints once it's constructed.
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