Substrate: Artcare acid-free foam.
That's a problem. DCO-Acrylic framing requires the spring-tension of batting to create the pressure that holds the item in place. Larger size requires more pressure; a greater thickness of batting. So, in the size you mentioned, foam board would not be rigid enough to resist deflection under the pressure. My suggestion would be to use a substrate of 3 mm or 4 mm aluminum composite material (ACM) such as ePanel, alucobond - there are many brands available. This is a very rigid, lightweight, gas-impermeable, preservation-worthy material and it should be suitable for this project.
Layer the ACM substrate with several thicknesses (usually 1/2") of traditional needle-punched polyester (not cotton) batting, then cover it with a suitable fabric - preferably one with some gripping action on the flag's fabric; perhaps a coarse weave.
Glazing: Reflection control UV filter acrylic.
Normally I would discourage using reflection control glazing, but since it would be in direct contact with the flag, its fuzzy visual effect would be minimized, and its microscopically-bumpy surface would reduce the appearance of abrasion over time. Good call.
Flag material: No clue. It's very thin and slick, and it has a little bit of a sheen to it. (Picture attached - hopefully it'll show enough detail).
No picture on my screen, but that's OK. The fabric probably would not be an issue.
Grommets: Negative. The pole side is a heavily reinforced sleeve (I'm sure there's a name for that, but I don't know what it is), so it has no metal to come into contact with the acrylic glazing.
Good. The added thickness of the pole-sleeve should be OK, as it would be pressed into the batting. If you use a flexible/stretchy fabric, it might deflect around the extra thickness without wrinkling.
The hems in the flag seem to be sturdy enough to tack it to the foam to give it a mechanical attachment, without sewing the whole perimeter.
I suggest allowing the flag to "float" on the padded fabric background without stitching at any points on its perimeter. That way, the flag can move freely; point-stitches might cause puckers/wrinkles over time.
The unfortunate thing is that this will be in a Masonic lodge room, which is not climate controlled most of the time. It'll go through lots of cold/hot swings over its lifetime. Is there something that I can do to mitigate the damage that the temperature/humidity changes will do?
Nothing I can think of. If I were you, I would tell them about the issues of unstable ambient conditions, the effects of expansion & contraction, and let them know that eventually, the acrylic may need to be replaced. It should be years - especially since you're using Reflection Control (non-glare) acrylic.