I mash in a cooler but I do the step mashes by using my boil kettle to do the steps and then transfer the mash into my cooler when I get to the final rest temp (pre-heat it first with some hot water so that you don't get a big temp drop).
Not sure what you mean when you say you can't heat your mash tun but if you can heat the water (unless you're all electric I guess), you can do the step and make it work.
I never thought to mash in my boil kettle initially. When I say I cant heat my tun i mean it is a cooler and i cant put it on a burner. What is the temp of your pre-heated ? Dont you lose a lot of temperature in the transfer? Do you overshoot your mash temperature to compensate for the transfer to the cooler?
I put hot tap water in the cooler while I'm doing the steps. I only do 5 gal batches of wit cause I don't drink it as much as other beers so when I hit 153 or so on my Thermapen, I just turn off the heat, dump out the water and dump the wort into the cooler (hot-side-aeration-a-phobes begin wails of disapproval now). Close the lid. Takes about 10 seconds total if you've got the cooler beside the kettle. Because the wort usually rises a bit anyway when I turn off the heat, it's probably over-shot slightly.
I take the temp after a few minutes in the cooler and if it's too hot I leave it open and stir a bit (or toss in a couple ice cubes). If it's way low, you could pull a decoction but for me I've always been more than close enough. I must admit, if the target temp is 152 and I get 150 or 153 etc, I really don't worry about it. Alternatively, you could have a pot of boiling water on the kitchen stove to do any fine-tuning of temp adjustments. Admittedly this could be more complicated in the winter than in the summer because of the larger temperature differences unless you have an indoor brewery. I usually make wit in the spring for summer drinking so it's not as much of an issue.
The only thing that is missing in this process is the mash out but I don't normally ever mash out. It could be helpful in this case cause of all the wheat but I use a good amount of rice hulls. If you really wanted to mash out, you could use your now empty boil kettle to boil some water and dump it in to raise the temp over 173F. The extra water shouldn't matter to the water:grain ratio because you're done mashing at this point.