It's just a face frame so all suggestions (TB2 - construction adhesive - biscuit jointer) should work. As an experiment, I once glued 2 scrap pieces
of bamboo flooring, finished side to finished side, using construction adhesive and they are bonded forever. TB2 would not glue those 2 pieces together because of the finish. But your working with clean bare wood surfaces so even TB2 should work.
The only problem with the construction adhesives (and Gorilla glue) is they expand so you'll need to deal with glue stained wood that will affect your finished product. No stain or finish will penetrate wood that has glue residue on it. But there are ways to deal with the problem...
Its' never hot or cold in New Hampshire... its' always seasonal.