This is an interesting thought. I have no experience with a batch sparge system like that, but I have heard of some experiment results that you may find interesting, and I can offer a process change you might find helpful.
First, an experiment was done on BBR where a listener suggested heating his initial mash infusion water along with the grains (which were seperated into a glass jar), in an effort to save time and have less of a differential between the room temp grains and hot strike water going into the mash. His results showed a minor time savings (literally 2 or 3 minutes), and he said hitting mash temperatures was a little easier. More importantly for you, slowly heating the grains along with the water did not seem to have any negative effects.
You idea is a little different, in that you will be heating grains that have already experienced a starch -> sugar conversion, because you are only concerned with trying this approach during the sparge. Honestly, I do not know if it will hurt your product. However, I have found that while I fly sparge, if I do not start with a mash out step, the sparge can be a bit of a struggle. I believe this is the case for two reasons. First, the mash-out makes the mash a bit thinner. Second, and this applies more to your situation, the additional infusion of heat keeps the sugars in the mash more liquified (think cold honey and warm honey) and easier to rinse. I do not know how heating/cooling/heating would effect the rinsing of the sugars. May have no effect; may have a big effect.
I understand you want to stick to two vessels, but does a bucket really count as a third vessel? If you only plan 1 batch sparge, I may be able to tweak the process a bit and eliminate your concern all together. However, a double batch sparge would make this process impossible.
Steps as follows.
1st - heat strike water in kettle.
2nd - infuse strike water and grains in mash tun (You mentioned a pump - good time to use it)
3rd - while mash is resting (maybe 30 minutes after strike), measure and add sparge water to kettle and slowly approach sparge temps (180/185/190 depending on ambient - go a little hot because some heat will be lost over a few minutes of sitting in a bucket). This is a timing step, so practice will make it easier.
4th - once mash is complete, and as sparge water approaches temp, begin/finish recirculating your mash for clarity (again, good use of the pump). This is a good time to think about emptying that sparge water into the bucket (ie- pseudo-3rd vessel). Again, some practice will be needed to get down the timing of this step. In reality, you only want the sparge water to be sitting in the bucket as long as it takes to empty the first runnings. Time is your enemy at this point.
5th- empty the mash tun into your now empty kettle as you would a typical batch sparge. Once an inch or two has been collected in the kettle, for more time savings, put a slow fire on the burner to get a jump on the boil. During this time, the first "batch sparge" is still sitting in the bucket around 175/180)
6th- as soon as the first runnings from the mash tun has stopped draining, carefully dump the sparge water from the bucket into the MLT. Stir, recirculate, and drain the second runnings into the kettle.
7th- boil, cool, pitch, ferment, condition, carbonate, and drink.
This method eliminates the concerns about taking sparge water and mashed grains from cold to hot, and will also save you a ton of time on brewday by multi-tasking. But again, it only works with a single batch sparge, and you have to be willing to make a big investment in a bucket.