1. There is nothing wrong with using a 26" MTB on the GDMBR. Yes, 29" wheels run more smoothy over a rough and washboarded trails, but the difference is gradual, not decisive.
2. More important is how to carry the gear. As a backpacking expert you probably know how to limit your load to essentials. Still the gear volume is probably too big for mounting all on handlebar, frame and saddle, like the Tour Divide racers do. Especially if you aim for 50 miles/day, you need more volume for food and liquids between the resupply points. This implies either panniers mounted on a rack or a trailer. I prefer panniers if the rack can be mounted with screw eyelets on the frame. I wouldn't trust clamping constructions for the rack, so if the frame has no screw eyelets I would go for a trailer.
3. I understand that the Specialized Hardrock is a hardtail with a basic suspension fork. This is OK, because the GD goes mostly over dirt roads and forest roads. There are no big jumps or intricate slaloms that require a high-end suspension fork. You will appreciate the suspension on the washboarded and rough roads of Wyoming and New Mexico. Half of the TD racers use a rigid fork, but this is because low weight and robustness is their prime objective. For you the prime objective should be to stay healthy on the bike, i.e. soften the impact on hands, joints and butt. However, a lockout on the front suspension is desirable to prevent it from absorbing pedalling power during climbing.
4. If your bike would be a full-suspension, you need a trailer because it is almost impossible to mount the rack and gear on a frame where tubes are flexing relative to each other. So I do not agree with MrBent on this point. I have never heard of or read about torquing destroying the bearings. However, I have experienced myself that torquing does reduce control over the bike on loose or sidewards-sloping surfaces, resulting in spills.
5. Improvement suggestion: fit the widest tires that your frame allows. The wider the tire, the lower the required pressure, the lower the imprint on a soft surface. Hence the lower the rolling resistance. Lower pressure also means more suspension, hence lower impact of shocks and ripples on body and wheels.
6. Improvement suggestion: Ergon-like grips.
Direction and season.
I have done the GD route in both directions and there are pros and cons for either choice. Based on my own experience and reading many blogs, I have constructed a chart. This is not rocket science, but a statistic of weather and road conditions over several years. I know that the Tour Divide race starts early June, mocking my N-to-S window, but hardships from snowed-in passes are part and parcel of this race. If you like to start in Banff and have some time-flexibility, I recommend to start between mid-July and early-August.