The pole barn company (considered a high end builder) that I used to sell for put 2x4 horizontal studs at 24" OC on the inside of the posts as you note you are planning to do. We used 24" wide R19 paper faced batts in the vertical direction in between these interior studs and the exterior studs - butted edge to edge tightly and stapled at the top. They were a nice tight fit and did not sag over time (evidenced by later remodels/repair work where we pulled down walls to find the insulation exactly where we left it).
The extra air space between the insulation and the exterior siding and interior wall covering provides a dead air space which further helps the insulation values of the wall structure.
Tip: Install the interior studs before you hang the insulation. It will take you a few sheets to get good at weaving the batts into place, but once you're used to it, it's nice to have the interior studs acting as another set of hands to hold the insulation.
Tip 2: If you know of locations where you plan to hang things on the walls (e.g. cabinets) place an extra horizontal stud there. Also, frame flush around openings (e.g. windows) so you don't have to hunt for the horizontal studs when installing trim later.
