Welcome to the board
There are 2 areas influenced by up and down movement (deflection) within a floor's construction. One is the span of the joists and the other is the plywood between the joists. The beams you've installed may have solved the deflection of the joists.
Was advised by an installer to just screw down 1/2" HardiBacker WITHOUT thinset over the OSB and WITHOUT 1/8" gaps. Claims 1/2" thickness will take care of any flatness issues (which are mostly under 1/8") and that gaps are not necessary.
Is he right?
Hell no! In fact your floor would fail and have no warranty. HardiBacker should be installed as they advise with a bed of mortar under their panels. Gaps and taping/mudding panel seams are also essential.
Also, the kitchen, front door, closet, and garage entrance are part of this install. Installer claims the HardiBacker acts as a moisture barrier.
Do I need to install a moisture barrier over the OSB? If so, what do you recommend?
HardiBacker isn't a moisture barrier ... and no moisture barrier is required.
If you have completely removed the flex of your joists, then you should reinforce the OSB to the joists ... sand the OSB seams if swollen or peaked and install the HardiBacker as HardiBacker recommends.
Dita membrane is another option instead of HardiBacker.
I'd recommend at least 1 control joint somewhere central of the floor within the 37' length.
