When I used to brew with a cooler mash tun I didn't preheat at all. Some ideas for next time:
1. Compensate for the heat capacity of your mash tun. For most vessels used in home brewing this can easily be compensated for by bumping your strike temp up by 2-3 degrees. Better too high than too low.
2. Rather than trying to hit a specific quantity of water and grain at a specific temperature, try to aim just for temperature. Get your strike water about 170F (likely about 5 degrees too hot), then dispense it to the mash tun until you about the the consistency you need. Stir well, wait 5 minutes, stir again, and take your reading. If you're low, add more hot water. If you're high, leave the cooler lid off and stir extra for a few minutes.
Stir well and at least every 15 minutes. There is significant temperature stratification between top and bottom of mash tun (i've seen 10 degrees in a 10G cooler).
3. Get a RIMS on your cooler. Then you can heat your exact volume of water directly in the cooler, which will preheat the mash tun automatically. This is basically what I do right now and its amazingly consistent. I always do 1.5 qt/lb and I always need my strike water +9F above the mash temp i want to hit.
4. Mash in your kettle if it's big enough. I used to do this too. Allows for step mashing. Only hassle is moving the mass of grain and water. I used to use a 2G bucket until it was light enough to pick up with a hot pad and dump. Made a huge sticky mess but very effective.