I think we radically overestimate how much traffic on the Interstate system is through traffic. Certainly long-haul trucking is important, and family vacations are a thing, but most of what travel you see is intraregional - I-22 is much more frequented by people going from Tupelo or New Albany or Holly Springs to go to Memphis or vice versa that it is for journeys between Little Rock and Atlanta - at that distance, business travelers are flying and most shipments would go by rail or air. The impact of I-269 on traffic levels on Lamar, even when complete, will likely be imperceptible.
On the last point, look at the maps here, specifically this figure from BTS. US 49 south of Jackson was actually carrying substantially more freight than I-22 in 2012; anecdotally, I-22 truck traffic doesn't seem that heavy even today (I regularly drive I-16 southeast of Macon and despite its reputation as a low-traffic interstate even it has more truck traffic on it than I usually see on I-22), and I'm not sure even the I-55 linkup will add that much more.
Despite the perception that trunk Interstates are there primarily to facilitate interregional travel, data repeatedly shows that there are significantly higher-traffic "nodes" extending out from the more densely populated areas that provide much of the aggregate traffic load for any given route. In many cases a look at usage of many Interstates demonstrates that one is not simply looking at one road or even one corridor -- but an effective series of "SIU's" linked together by the lesser-utilized segments. In the case of I-22, one such node is the Olive Branch/Byhalia area, where much of the traffic is simply folks getting on one exit and getting off a few exits later -- it's a convenient roadway for local usage; the fact that it's a full-fledged trunk Interstate is simply "icing on the cake", so to speak. Once into the woods, things thin out a bit until around New Albany-Tupelo, which provides an additional traffic node (enhanced in the last few years by the Toyota plant near MS 9 west of Tupelo). That too thins out toward the Alabama state line; with a bit of sparsity before the penultimate node (Winfield-Jasper) occurs, with the resultant dominance of local traffic;
that node blends with the final Birmingham segment. The fact that there is a modicum of through traffic doesn't affect the fact that right now I-22 functions as much as a chain of local "SIU's" as a single interregional entity.
But that route is still very much in its teething stages; except for the Toyota plant, which ironically ships most of its product out by rail (although trucks carrying "just-in-time" parts to the facility
do account for a good piece of I-22 commercial traffic), commercial development along the 205-mile overall corridor just hasn't happened to date. What intrinsic Birmingham-Memphis truck traffic there is uses it, of course -- just as they would have used US 78 before the freeway was constructed; it's the logical routing. But now that it connects to the rest of the Interstate network (albeit convolutedly at its west end for the time being), one might expect gradual increases in overall usage as interregional use expands. However, local traffic will also increase, particularly along "linear" nodes strung out with I-22 as the focal point -- such as the Winfield/Jasper segment mentioned earlier. Expect localized traffic to dominate the overall picture on this relatively new corridor until interregional traffic patterns begin shifting to take advantage of its presence -- which should take at least another 10 years or so; as connections (I-269 west to I-55/69) are made and the corridor segments beyond the I-22 connection itself (including the long-delayed Crump Blvd. improvement on I-55 itself), I-22 will present itself as a logical option in the SE>NW regional picture. Do note that the commercial-traffic map cited above is based upon the facility prior to its MS improvements and also prior to its formal designation (likely figures from 2010 or earlier).