Will the hot thermal mass counter my attempts of cooling the house down?
Yes!
There is some good research that looks into all this so there is no need to reinvent the wheel unless you just dont believe its true. The only type of climate where a purely thermal mass wall (no insulation) works are high elevation deserts with wide diurnal temp swings yet fairly consistent temps throughout the year. Does such a climate even exist anymore? ORNL, who I think has done the main body of research suggests Phoenix AZ and Bakersfield CA as examples but even there I have my doubts.
Outside of those climates, thermal mass
should be completely inside the building envelope (air and thermal barriers) to have any benefit. Its very debatable if adding thermal mass beyond what is needed anyway is cost-effective.
In South Texas (US building climate zone 2) the temps wont cool off enough at night to swing the temps back to your favor for cooling. Even your coolest listed nights of 76 wont cut it. Your house will be much more comfortable, cheaper to condition, and more durable if you follow the IECC 2012 table 402.2 for prescriptive R values. This is the MINIMUM for most concerned with building performance. More R= more comfort, less
energy costs.
This current energy code calls for a blower door test of at least 5 ACH50 in your climate zone. This requirement is sure to be lowered in the upcoming 2015 version and I would shoot for an ACH50 of 1.5 or below. All other things being average, this is the cheapest and most effective path to staying comfortable and having low energy/environmental costs. Less air infiltration= more comfort, less energy costs. The grand caveat of this paragraph is that below 5 ACH50 most experts think you should have a mechanical system for fresh air introduction. I think if youre concerned about Indoor Air Quality, you should have that anyway.
Along with this "modern building science" advice, take some tips from the old timers that didnt have AC: Mass the shape longer from East to West to minimize surface area in those directions and avoid glazing (windows) on those elevations if they arent shaded by a big porch (still avoid any west windows). Landscape to shade these walls and roof if possible. Build big overhangs with highly reflective roofing. Roof insulation is your friend